graphing assignment 2

Graphing Practice for Secondary Science

At the beginning of the year, it’s good to review graphing and make your expectations clear on what you expect when students turn in a graph.

• Clearly labeled X and Y axes?
• Proper scale?
• Nearly drawn bars?

If you teach physical science, this is pretty critical.

Here is a round-up of graphing resources you can use with students:

1. Graphing Poster: A great way to reinforce your graphing expectations is to have them visible for students. This graphing checklist can be placed in student binders or hung on the wall for students to reference when they work on a graph.

2. Turner’s Graph of the Week: This stellar website has a weekly graphing worksheet that is sure to engage students. Graphing topics are timely, relevant, and engaging.

3. Graphing stories: This website has video clips students watch, analyze, and create a graph from. (Most are motion related, so this is a great site for physical science teachers).

4. Smart Graphs: This digital activity has students read through a scenario and decide which is the best type of graph to represent events in the story.

5. ACS: Here are a series of graphing activities from American Chemical Society.

6. Graphing Stations: This station activity has 8 stations students rotate through that all relate to graphing. Activities include identifying which type of graph to use, creating titles for graphs, watching a video clip, and arranging a jigsaw puzzle.

7. Graphing Analysis: Here is a freebie on TpT from Amy Brown Science.

8. What’s going on in this graph? In this series from the New York Times, students check out graphs that are published weekly. Ask your students- what do you notice? What do you wonder? What’s going on?

9. Create a Graph: Want students to practice creating their own graphs digitally? If you don’t think your students are ready to tackle Excel, try out this user friendly website .

10. How to Spot a Misleading Graph: This TEd-Ex video shows students how graphs can mislead viewers. It brings up great discussion points!

I hope you find those useful!

• Read more about: Literacy

Hi, I'm Becca!

Search the site, browse by category.

• A list of ALL blog posts
• Back to School
• Biochemistry
• Body Systems
• Classification
• Classroom Decor
• Classroom Management
• Distance Learning
• End of the School Year
• Experiments
• Field Trips
• For NEW Teachers
• Formative Assessment
• Media in the Classroom
• Microscopes
• Photosynthesis & Respiration
• Plate Tectonics
• Sustainability
• Teacher Tips
• Weather and Climate

Get Freebies!

You might also like....

Read Aloud Closure Assignments

Biology Picture Book Recommendations

Earth and Space Science Picture Book Recommendations

Want a fun way to practice science vocabulary? Try out seek and finds!

Privacy Overview

Free Printable Math Worksheets for Algebra 2

Created with infinite algebra 2, stop searching. create the worksheets you need with infinite algebra 2..

• Fast and easy to use
• Multiple-choice & free-response
• Never runs out of questions
• Multiple-version printing

Free 14-Day Trial

• Order of operations
• Evaluating expressions
• Simplifying algebraic expressions
• Multi-step equations
• Work word problems
• Distance-rate-time word problems
• Mixture word problems
• Absolute value equations
• Multi-step inequalities
• Compound inequalities
• Absolute value inequalities
• Discrete relations
• Continuous relations
• Evaluating and graphing functions
• Review of linear equations
• Graphing absolute value functions
• Graphing linear inequalities
• Direct and inverse variation
• Systems of two linear inequalities
• Systems of two equations
• Systems of two equations, word problems
• Points in three dimensions
• Systems of three equations, elimination
• Systems of three equations, substitution
• Basic matrix operations
• Matrix multiplication
• All matrix operations combined
• Matrix inverses
• Geometric transformations with matrices
• Operations with complex numbers
• Properties of complex numbers
• Rationalizing imaginary denominators
• Properties of parabolas
• Vertex form
• Graphing quadratic inequalities
• Factoring quadratic expressions
• Solving quadratic equations w/ square roots
• Solving quadratic equations by factoring
• Completing the square
• Solving equations by completing the square
• Solving equations with the quadratic formula
• The discriminant
• Naming and simple operations
• Factoring a sum/difference of cubes
• Factoring by grouping
• Factoring quadratic form
• Factoring using all techniques
• Factors and Zeros
• The Remainder Theorem
• Irrational and Imaginary Root Theorems
• Descartes' Rule of Signs
• More on factors, zeros, and dividing
• The Rational Root Theorem
• Polynomial equations
• Basic shape of graphs of polynomials
• Graphing polynomial functions
• The Binomial Theorem
• Evaluating functions
• Function operations
• Inverse functions
• Simplifying radicals
• Operations with radical expressions
• Dividing radical expressions
• Radicals and rational exponents
• Simplifying rational exponents
• Square root equations
• Rational exponent equations
• Graphing radicals
• Graphing & properties of parabolas
• Equations of parabolas
• Graphing & properties of circles
• Equations of circles
• Graphing & properties of ellipses
• Equations of ellipses
• Graphing & properties of hyperbolas
• Equations of hyperbolas
• Classifying conic sections
• Eccentricity
• Systems of quadratic equations
• Graphing simple rational functions
• Graphing general rational functions
• Simplifying rational expressions
• Multiplying / dividing rational expressions
• Adding / subtracting rational expressions
• Complex fractions
• Solving rational equations
• The meaning of logarithms
• Properties of logarithms
• The change of base formula
• Writing logs in terms of others
• Logarithmic equations
• Inverse functions and logarithms
• Exponential equations not requiring logarithms
• Exponential equations requiring logarithms
• Graphing logarithms
• Graphing exponential functions
• Discrete exponential growth and decay word problems
• Continuous exponential growth and decay word problems
• General sequences
• Arithmetic sequences
• Geometric sequences
• Comparing Arithmetic/Geometric Sequences
• General series
• Arithmetic series
• Arithmetic/Geometric Means w/ Sequences
• Finite geometric series
• Infinite geometric series
• Right triangle trig: Evaluating ratios
• Right triangle trig: Missing sides/angles
• Angles and angle measure
• Co-terminal angles and reference angles
• Arc length and sector area
• Trig ratios of general angles
• Exact trig ratios of important angles
• The Law of Sines
• The Law of Cosines
• Graphing trig functions
• Translating trig functions
• Angle Sum/Difference Identities
• Double-/Half-Angle Identities
• Sample spaces and The Counting Principle
• Independent and dependent events
• Mutualy exclusive events
• Permutations
• Combinations
• Permutations vs combinations
• Probability using permutations and combinations

Number Sense

Understanding numbers, their relationships and numerical reasoning

Using symbols to solve equations and express patterns

Studying shapes, sizes and spatial relationships in mathematics

Measurement

Quantifying and comparing attributes like length, weight and volume

Performing mathematical operations like addition, subtraction, division

Probability and Statistics

Analyzing uncertainty and likelihood of events and outcomes

Calculator Suite

Exploring functions, solving equations, constructing geometric shapes

Graphing Calculator

Visualizing equations and functions with interactive graphs and plots

Exploring geometric concepts and constructions in a dynamic environment

3D Calculator

Graphing functions and performing calculations in 3D

Scientific Calculator

Performing calculations with fractions, statistics and exponential functions

Exploring apps bundle including free tools for geometry, spreadsheet and CAS

Teach and learn math in a smarter way

GeoGebra is more than a set of free tools to do math. It’s a platform to connect enthusiastic teachers and students and offer them a new way to explore and learn about math.

Recommended math resources:

GeoGebra-curated for Grades 4 to 8

What we offer

Math Resources

Our newest collection of GeoGebra Math Resources has been meticulously crafted by our expert team for Grades 4 to 8.

Math Calculators & Apps

Free tools for an interactive learning and exam experience. Available on all platforms.

Classroom Collaboration

Interactive math lessons with GeoGebra materials available. Integration with various LMSs supported.

Math Practice

Get support in step-by-step math exercises, explore different solution paths and build confidence in solving algebraic problems.

Trusted by millions of teachers and students worldwide

GeoGebra allowed us to provide access and equity to our diverse student population.

Irina Kimyagarov

Teacher, New York, US

I appreciate the responsiveness of the GeoGebra staff and the larger community.

Nancy Swarat

Teacher / OCTM, Oregon, US

GeoGebra has changed the way I teach.

Maria Meyer

Teacher, New Mexico, US

GeoGebra brings mathematics to life!

Steven C. Silvestri

My students LOVE using GeoGebra.

Peter Myers

Teacher, New York City, US

Teachers & Students

Make math interactive.

Try our exploration activities to discover important math concepts, then use our practice activities to master these skills.

Explore math with free calculators

Explore our easy-to-use calculators that can be used to promote student-centered discovery-based learning. Perform calculations for any level of mathematics including 3D.

Engage every student

Our Classroom learning platform allows teachers to view student progress real-time and provide individual feedback for personalized learning experience. It helps teachers to encourage active participation and discussion.

Solve problems step-by-step

Our Math Practice tool offers new ways for learners to access algebraic transformation in an understandable way. Let your students build comfort and fluency in solving algebraic problems, like simplifying algebraic expressions or solving linear equations, while getting instant hints and feedback.

Our mission

Our mission is to give the best tools for teachers to empower their students to unleash their greatest potential. We go beyond being just a collection of tools. Striving to connect passionate individuals from the education world, we offer a fresh approach to teaching, exploring, and learning mathematics.

Get started using GeoGebra today

Create a free account so you can save your progress any time and access thousands of math resources for you to customize and share with others

3.2 Domain and Range

Learning objectives.

In this section, you will:

• Find the domain of a function defined by an equation.
• Graph piecewise-defined functions.

Horror and thriller movies are both popular and, very often, extremely profitable. When big-budget actors, shooting locations, and special effects are included, however, studios count on even more viewership to be successful. Consider five major thriller/horror entries from the early 2000s— I am Legend , Hannibal , The Ring , The Grudge , and The Conjuring . Figure 1 shows the amount, in dollars, each of those movies grossed when they were released as well as the ticket sales for horror movies in general by year. Notice that we can use the data to create a function of the amount each movie earned or the total ticket sales for all horror movies by year. In creating various functions using the data, we can identify different independent and dependent variables, and we can analyze the data and the functions to determine the domain and range. In this section, we will investigate methods for determining the domain and range of functions such as these.

Finding the Domain of a Function Defined by an Equation

In Functions and Function Notation , we were introduced to the concepts of domain and range . In this section, we will practice determining domains and ranges for specific functions. Keep in mind that, in determining domains and ranges, we need to consider what is physically possible or meaningful in real-world examples, such as tickets sales and year in the horror movie example above. We also need to consider what is mathematically permitted. For example, we cannot include any input value that leads us to take an even root of a negative number if the domain and range consist of real numbers. Or in a function expressed as a formula, we cannot include any input value in the domain that would lead us to divide by 0.

We can visualize the domain as a “holding area” that contains “raw materials” for a “function machine” and the range as another “holding area” for the machine’s products. See Figure 2 .

We can write the domain and range in interval notation , which uses values within brackets to describe a set of numbers. In interval notation, we use a square bracket [ when the set includes the endpoint and a parenthesis ( to indicate that the endpoint is either not included or the interval is unbounded. For example, if a person has $100 to spend, they would need to express the interval that is more than 0 and less than or equal to 100 and write ( 0 , 100 ] . ( 0 , 100 ] . We will discuss interval notation in greater detail later.

Let’s turn our attention to finding the domain of a function whose equation is provided. Oftentimes, finding the domain of such functions involves remembering three different forms. First, if the function has no denominator or an odd root, consider whether the domain could be all real numbers. Second, if there is a denominator in the function’s equation, exclude values in the domain that force the denominator to be zero. Third, if there is an even root, consider excluding values that would make the radicand negative.

Before we begin, let us review the conventions of interval notation:

• The smallest number from the interval is written first.
• The largest number in the interval is written second, following a comma.
• Parentheses, ( or ), are used to signify that an endpoint value is not included, called exclusive.
• Brackets, [ or ], are used to indicate that an endpoint value is included, called inclusive.

See Figure 3 for a summary of interval notation.

Finding the Domain of a Function as a Set of Ordered Pairs

Find the domain of the following function: { ( 2 , 10 ) , ( 3 , 10 ) , ( 4 , 20 ) , ( 5 , 30 ) , ( 6 , 40 ) } { ( 2 , 10 ) , ( 3 , 10 ) , ( 4 , 20 ) , ( 5 , 30 ) , ( 6 , 40 ) } .

First identify the input values. The input value is the first coordinate in an ordered pair . There are no restrictions, as the ordered pairs are simply listed. The domain is the set of the first coordinates of the ordered pairs.

Find the domain of the function:

{ ( −5 , 4 ) , ( 0 , 0 ) , ( 5 , −4 ) , ( 10 , −8 ) , ( 15 , −12 ) } { ( −5 , 4 ) , ( 0 , 0 ) , ( 5 , −4 ) , ( 10 , −8 ) , ( 15 , −12 ) }

Given a function written in equation form, find the domain.

• Identify the input values.
• Identify any restrictions on the input and exclude those values from the domain.
• Write the domain in interval form, if possible.

Finding the Domain of a Function

Find the domain of the function f ( x ) = x 2 − 1. f ( x ) = x 2 − 1.

The input value, shown by the variable x x in the equation, is squared and then the result is lowered by one. Any real number may be squared and then be lowered by one, so there are no restrictions on the domain of this function. The domain is the set of real numbers.

In interval form, the domain of f f is ( − ∞ , ∞ ) . ( − ∞ , ∞ ) .

Find the domain of the function: f ( x ) = 5 − x + x 3 . f ( x ) = 5 − x + x 3 .

Given a function written in an equation form that includes a fraction, find the domain.

• Identify any restrictions on the input. If there is a denominator in the function’s formula, set the denominator equal to zero and solve for x x . If the function’s formula contains an even root, set the radicand greater than or equal to 0, and then solve.
• Write the domain in interval form, making sure to exclude any restricted values from the domain.

Finding the Domain of a Function Involving a Denominator

Find the domain of the function f ( x ) = x + 1 2 − x . f ( x ) = x + 1 2 − x .

When there is a denominator, we want to include only values of the input that do not force the denominator to be zero. So, we will set the denominator equal to 0 and solve for x . x .

Now, we will exclude 2 from the domain. The answers are all real numbers where x < 2 x < 2 or x > 2 x > 2 as shown in Figure 4 . We can use a symbol known as the union, ∪ , ∪ , to combine the two sets. In interval notation, we write the solution: ( −∞ , 2 ) ∪ ( 2 , ∞ ) . ( −∞ , 2 ) ∪ ( 2 , ∞ ) .

Find the domain of the function: f ( x ) = 1 + 4 x 2 x − 1 . f ( x ) = 1 + 4 x 2 x − 1 .

Given a function written in equation form including an even root, find the domain.

• Since there is an even root, exclude any real numbers that result in a negative number in the radicand. Set the radicand greater than or equal to zero and solve for x . x .
• The solution(s) are the domain of the function. If possible, write the answer in interval form.

Finding the Domain of a Function with an Even Root

Find the domain of the function f ( x ) = 7 − x . f ( x ) = 7 − x .

When there is an even root in the formula, we exclude any real numbers that result in a negative number in the radicand.

Set the radicand greater than or equal to zero and solve for x . x .

Now, we will exclude any number greater than 7 from the domain. The answers are all real numbers less than or equal to 7 , 7 , or ( − ∞ , 7 ] . ( − ∞ , 7 ] .

Find the domain of the function f ( x ) = 5 + 2 x . f ( x ) = 5 + 2 x .

Can there be functions in which the domain and range do not intersect at all?

Yes. For example, the function f ( x ) = − 1 x f ( x ) = − 1 x has the set of all positive real numbers as its domain but the set of all negative real numbers as its range. As a more extreme example, a function’s inputs and outputs can be completely different categories (for example, names of weekdays as inputs and numbers as outputs, as on an attendance chart), in such cases the domain and range have no elements in common.

Using Notations to Specify Domain and Range

In the previous examples, we used inequalities and lists to describe the domain of functions. We can also use inequalities, or other statements that might define sets of values or data, to describe the behavior of the variable in set-builder notation . For example, { x | 10 ≤ x < 30 } { x | 10 ≤ x < 30 } describes the behavior of x x in set-builder notation. The braces { } { } are read as “the set of,” and the vertical bar | is read as “such that,” so we would read { x | 10 ≤ x < 30 } { x | 10 ≤ x < 30 } as “the set of x -values such that 10 is less than or equal to x , x , and x x is less than 30.”

Figure 5 compares inequality notation, set-builder notation, and interval notation.

To combine two intervals using inequality notation or set-builder notation, we use the word “or.” As we saw in earlier examples, we use the union symbol, ∪ , ∪ , to combine two unconnected intervals. For example, the union of the sets { 2 , 3 , 5 } { 2 , 3 , 5 } and { 4 , 6 } { 4 , 6 } is the set { 2 , 3 , 4 , 5 , 6 } . { 2 , 3 , 4 , 5 , 6 } . It is the set of all elements that belong to one or the other (or both) of the original two sets. For sets with a finite number of elements like these, the elements do not have to be listed in ascending order of numerical value. If the original two sets have some elements in common, those elements should be listed only once in the union set. For sets of real numbers on intervals, another example of a union is

Set-Builder Notation and Interval Notation

Set-builder notation is a method of specifying a set of elements that satisfy a certain condition. It takes the form { x | statement about  x } { x | statement about  x } which is read as, “the set of all x x such that the statement about x x is true.” For example,

Interval notation is a way of describing sets that include all real numbers between a lower limit that may or may not be included and an upper limit that may or may not be included. The endpoint values are listed between brackets or parentheses. A square bracket indicates inclusion in the set, and a parenthesis indicates exclusion from the set. For example,

Given a line graph, describe the set of values using interval notation.

• Identify the intervals to be included in the set by determining where the heavy line overlays the real line.
• At the left end of each interval, use [ with each end value to be included in the set (solid dot) or ( for each excluded end value (open dot).
• At the right end of each interval, use ] with each end value to be included in the set (filled dot) or ) for each excluded end value (open dot).
• Use the union symbol ∪ ∪ to combine all intervals into one set.

Describing Sets on the Real-Number Line

Describe the intervals of values shown in Figure 6 using inequality notation, set-builder notation, and interval notation.

To describe the values, x , x , included in the intervals shown, we would say, “ x x is a real number greater than or equal to 1 and less than or equal to 3, or a real number greater than 5.”

Remember that, when writing or reading interval notation, using a square bracket means the boundary is included in the set. Using a parenthesis means the boundary is not included in the set.

Given Figure 7 , specify the graphed set in

• ⓑ set-builder notation
• ⓒ interval notation

Finding Domain and Range from Graphs

Another way to identify the domain and range of functions is by using graphs. Because the domain refers to the set of possible input values, the domain of a graph consists of all the input values shown on the x -axis. The range is the set of possible output values, which are shown on the y -axis. Keep in mind that if the graph continues beyond the portion of the graph we can see, the domain and range may be greater than the visible values. See Figure 8 .

We can observe that the graph extends horizontally from −5 −5 to the right without bound, so the domain is [ −5 , ∞ ) . [ −5 , ∞ ) . The vertical extent of the graph is all range values 5 5 and below, so the range is ( −∞ , 5 ] . ( −∞ , 5 ] . Note that the domain and range are always written from smaller to larger values, or from left to right for domain, and from the bottom of the graph to the top of the graph for range.

Finding Domain and Range from a Graph

Find the domain and range of the function f f whose graph is shown in Figure 9 .

We can observe that the horizontal extent of the graph is –3 to 1, so the domain of f f is ( − 3 , 1 ] . ( − 3 , 1 ] .

The vertical extent of the graph is 0 to –4, so the range is [ − 4 , 0 ] . [ − 4 , 0 ] . See Figure 10 .

Finding Domain and Range from a Graph of Oil Production

Find the domain and range of the function f f whose graph is shown in Figure 11 .

The input quantity along the horizontal axis is “years,” which we represent with the variable t t for time. The output quantity is “thousands of barrels of oil per day,” which we represent with the variable b b for barrels. The graph may continue to the left and right beyond what is viewed, but based on the portion of the graph that is visible, we can determine the domain as 1973 ≤ t ≤ 2008 1973 ≤ t ≤ 2008 and the range as approximately 180 ≤ b ≤ 2010. 180 ≤ b ≤ 2010.

In interval notation, the domain is [1973, 2008], and the range is about [180, 2010]. For the domain and the range, we approximate the smallest and largest values since they do not fall exactly on the grid lines.

Given Figure 12 , identify the domain and range using interval notation.

Can a function’s domain and range be the same?

Yes. For example, the domain and range of the cube root function are both the set of all real numbers.

Finding Domains and Ranges of the Toolkit Functions

We will now return to our set of toolkit functions to determine the domain and range of each.

Given the formula for a function, determine the domain and range.

• Exclude from the domain any input values that result in division by zero.
• Exclude from the domain any input values that have nonreal (or undefined) number outputs.
• Use the valid input values to determine the range of the output values.
• Look at the function graph and table values to confirm the actual function behavior.

Finding the Domain and Range Using Toolkit Functions

Find the domain and range of f ( x ) = 2 x 3 − x . f ( x ) = 2 x 3 − x .

There are no restrictions on the domain, as any real number may be cubed and then subtracted from the result.

The domain is ( − ∞ , ∞ ) ( − ∞ , ∞ ) and the range is also ( − ∞ , ∞ ) . ( − ∞ , ∞ ) .

Finding the Domain and Range

Find the domain and range of f ( x ) = 2 x + 1 . f ( x ) = 2 x + 1 .

We cannot evaluate the function at −1 −1 because division by zero is undefined. The domain is ( − ∞ , −1 ) ∪ ( −1 , ∞ ) . ( − ∞ , −1 ) ∪ ( −1 , ∞ ) . Because the function is never zero, we exclude 0 from the range. The range is ( − ∞ , 0 ) ∪ ( 0 , ∞ ) . ( − ∞ , 0 ) ∪ ( 0 , ∞ ) .

Find the domain and range of f ( x ) = 2 x + 4 . f ( x ) = 2 x + 4 .

We cannot take the square root of a negative number, so the value inside the radical must be nonnegative.

The domain of f ( x ) f ( x ) is [ − 4 , ∞ ) . [ − 4 , ∞ ) .

We then find the range. We know that f ( − 4 ) = 0 , f ( − 4 ) = 0 , and the function value increases as x x increases without any upper limit. We conclude that the range of f f is [ 0 , ∞ ) . [ 0 , ∞ ) .

Figure 22 represents the function f . f .

Find the domain and range of f ( x ) = − 2 − x . f ( x ) = − 2 − x .

Graphing Piecewise-Defined Functions

Sometimes, we come across a function that requires more than one formula in order to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function f ( x ) = | x | . f ( x ) = | x | . With a domain of all real numbers and a range of values greater than or equal to 0, absolute value can be defined as the magnitude , or modulus , of a real number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to be greater than or equal to 0.

If we input 0, or a positive value, the output is the same as the input.

If we input a negative value, the output is the opposite of the input.

Because this requires two different processes or pieces, the absolute value function is an example of a piecewise function. A piecewise function is a function in which more than one formula is used to define the output over different pieces of the domain.

We use piecewise functions to describe situations in which a rule or relationship changes as the input value crosses certain “boundaries.” For example, we often encounter situations in business for which the cost per piece of a certain item is discounted once the number ordered exceeds a certain value. Tax brackets are another real-world example of piecewise functions. For example, consider a simple tax system in which incomes up to $10,000 are taxed at 10%, and any additional income is taxed at 20%. The tax on a total income S S would be 0.1 S 0.1 S if S ≤ $ 10 , 000 S ≤ $ 10 , 000 and $ 1000 + 0.2 ( S − $ 10 , 000 ) $ 1000 + 0.2 ( S − $ 10 , 000 ) if S > $ 10 , 000. S > $ 10 , 000.

Piecewise Function

A piecewise function is a function in which more than one formula is used to define the output. Each formula has its own domain, and the domain of the function is the union of all these smaller domains. We notate this idea like this:

In piecewise notation, the absolute value function is

Given a piecewise function, write the formula and identify the domain for each interval.

• Identify the intervals for which different rules apply.
• Determine formulas that describe how to calculate an output from an input in each interval.
• Use braces and if-statements to write the function.

Writing a Piecewise Function

A museum charges $5 per person for a guided tour with a group of 1 to 9 people or a fixed $50 fee for a group of 10 or more people. Write a function relating the number of people, n , n , to the cost, C . C .

Two different formulas will be needed. For n -values under 10, C = 5 n . C = 5 n . For values of n n that are 10 or greater, C = 50. C = 50.

The function is represented in Figure 23 . The graph is a diagonal line from n = 0 n = 0 to n = 10 n = 10 and a constant after that. In this example, the two formulas agree at the meeting point where n = 10 , n = 10 , but not all piecewise functions have this property.

Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, C , C , in dollars for g g gigabytes of data transfer.

Find the cost of using 1.5 gigabytes of data and the cost of using 4 gigabytes of data.

To find the cost of using 1.5 gigabytes of data, C ( 1.5 ) , C ( 1.5 ) , we first look to see which part of the domain our input falls in. Because 1.5 is less than 2, we use the first formula.

To find the cost of using 4 gigabytes of data, C ( 4 ) , C ( 4 ) , we see that our input of 4 is greater than 2, so we use the second formula.

The function is represented in Figure 24 . We can see where the function changes from a constant to a shifted and stretched identity at g = 2. g = 2. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

Given a piecewise function, sketch a graph.

• Indicate on the x -axis the boundaries defined by the intervals on each piece of the domain.
• For each piece of the domain, graph on that interval using the corresponding equation pertaining to that piece. Do not graph two functions over one interval because it would violate the criteria of a function.

Graphing a Piecewise Function

Sketch a graph of the function.

Each of the component functions is from our library of toolkit functions, so we know their shapes. We can imagine graphing each function and then limiting the graph to the indicated domain. At the endpoints of the domain, we draw open circles to indicate where the endpoint is not included because of a less-than or greater-than inequality; we draw a closed circle where the endpoint is included because of a less-than-or-equal-to or greater-than-or-equal-to inequality.

Figure 25 shows the three components of the piecewise function graphed on separate coordinate systems.

Now that we have sketched each piece individually, we combine them in the same coordinate plane. See Figure 26 .

Note that the graph does pass the vertical line test even at x = 1 x = 1 and x = 2 x = 2 because the points ( 1 , 3 ) ( 1 , 3 ) and ( 2 , 2 ) ( 2 , 2 ) are not part of the graph of the function, though ( 1 , 1 ) ( 1 , 1 ) and ( 2 , 3 ) ( 2 , 3 ) are.

Graph the following piecewise function.

Can more than one formula from a piecewise function be applied to a value in the domain?

No. Each value corresponds to one equation in a piecewise formula.

Access these online resources for additional instruction and practice with domain and range.

• Domain and Range of Square Root Functions
• Determining Domain and Range
• Find Domain and Range Given the Graph
• Find Domain and Range Given a Table
• Find Domain and Range Given Points on a Coordinate Plane

3.2 Section Exercises

Why does the domain differ for different functions?

How do we determine the domain of a function defined by an equation?

Explain why the domain of f ( x ) = x 3 f ( x ) = x 3 is different from the domain of f ( x ) = x . f ( x ) = x .

When describing sets of numbers using interval notation, when do you use a parenthesis and when do you use a bracket?

How do you graph a piecewise function?

For the following exercises, find the domain of each function using interval notation.

f ( x ) = − 2 x ( x − 1 ) ( x − 2 ) f ( x ) = − 2 x ( x − 1 ) ( x − 2 )

f ( x ) = 5 − 2 x 2 f ( x ) = 5 − 2 x 2

f ( x ) = 3 x − 2 f ( x ) = 3 x − 2

f ( x ) = 3 − 6 − 2 x f ( x ) = 3 − 6 − 2 x

f ( x ) = 4 − 3 x f ( x ) = 4 − 3 x

f ( x ) = x 2 + 4 f ( x ) = x 2 + 4

f ( x ) = 1 − 2 x 3 f ( x ) = 1 − 2 x 3

f ( x ) = x − 1 3 f ( x ) = x − 1 3

f ( x ) = 9 x − 6 f ( x ) = 9 x − 6

f ( x ) = 3 x + 1 4 x + 2 f ( x ) = 3 x + 1 4 x + 2

f ( x ) = x + 4 x − 4 f ( x ) = x + 4 x − 4

f ( x ) = x − 3 x 2 + 9 x − 22 f ( x ) = x − 3 x 2 + 9 x − 22

f ( x ) = 1 x 2 − x − 6 f ( x ) = 1 x 2 − x − 6

f ( x ) = 2 x 3 − 250 x 2 − 2 x − 15 f ( x ) = 2 x 3 − 250 x 2 − 2 x − 15

f ( x ) = 5 x − 3 f ( x ) = 5 x − 3

f ( x ) = 2 x + 1 5 − x f ( x ) = 2 x + 1 5 − x

f ( x ) = x − 4 x − 6 f ( x ) = x − 4 x − 6

f ( x ) = x − 6 x − 4 f ( x ) = x − 6 x − 4

f ( x ) = x x f ( x ) = x x

f ( x ) = x 2 − 9 x x 2 − 81 f ( x ) = x 2 − 9 x x 2 − 81

Find the domain of the function f ( x ) = 2 x 3 − 50 x f ( x ) = 2 x 3 − 50 x by:

• ⓐ using algebra.
• ⓑ graphing the function in the radicand and determining intervals on the x -axis for which the radicand is nonnegative.

For the following exercises, write the domain and range of each function using interval notation.

For the following exercises, sketch a graph of the piecewise function. Write the domain in interval notation.

f ( x ) = { x + 1 if x < − 2 − 2 x − 3 if x ≥ − 2 f ( x ) = { x + 1 if x < − 2 − 2 x − 3 if x ≥ − 2

f ( x ) = { 2 x − 1 if x < 1 1 + x if x ≥ 1 f ( x ) = { 2 x − 1 if x < 1 1 + x if x ≥ 1

f ( x ) = { x + 1 if x < 0 x − 1 if x > 0 f ( x ) = { x + 1 if x < 0 x − 1 if x > 0

f ( x ) = { 3 if x < 0 x if x ≥ 0 f ( x ) = { 3 if x < 0 x if x ≥ 0

f ( x ) = { x 2      if  x < 0 1 − x  if  x > 0 f ( x ) = { x 2      if  x < 0 1 − x  if  x > 0

f ( x ) = { x 2 x + 2 if x < 0 if x ≥ 0 f ( x ) = { x 2 x + 2 if x < 0 if x ≥ 0

f ( x ) = { x + 1 if x < 1 x 3 if x ≥ 1 f ( x ) = { x + 1 if x < 1 x 3 if x ≥ 1

f ( x ) = { | x | 1 if x < 2 if x ≥ 2 f ( x ) = { | x | 1 if x < 2 if x ≥ 2

For the following exercises, given each function f , f , evaluate f ( −3 ) , f ( −2 ) , f ( −1 ) , f ( −3 ) , f ( −2 ) , f ( −1 ) , and f ( 0 ) . f ( 0 ) .

f ( x ) = { 1 if  x ≤ − 3 0 if  x > − 3 f ( x ) = { 1 if  x ≤ − 3 0 if  x > − 3

f ( x ) = { − 2 x 2 + 3 if  x ≤ − 1 5 x − 7 if  x > − 1 f ( x ) = { − 2 x 2 + 3 if  x ≤ − 1 5 x − 7 if  x > − 1

For the following exercises, given each function f , f , evaluate f ( −1 ) , f ( 0 ) , f ( 2 ) , f ( −1 ) , f ( 0 ) , f ( 2 ) , and f ( 4 ) . f ( 4 ) .

f ( x ) = { 7 x + 3 if x < 0 7 x + 6 if x ≥ 0 f ( x ) = { 7 x + 3 if x < 0 7 x + 6 if x ≥ 0

f ( x ) = { x 2 − 2 if x < 2 4 + | x − 5 | if x ≥ 2 f ( x ) = { x 2 − 2 if x < 2 4 + | x − 5 | if x ≥ 2

f ( x ) = { 5 x if x < 0 3 if 0 ≤ x ≤ 3 x 2 if x > 3 f ( x ) = { 5 x if x < 0 3 if 0 ≤ x ≤ 3 x 2 if x > 3

For the following exercises, write the domain for the piecewise function in interval notation.

f ( x ) = { x 2 − 2 if x < 1 − x 2 + 2 if x > 1 f ( x ) = { x 2 − 2 if x < 1 − x 2 + 2 if x > 1

f ( x ) = { 2 x − 3 − 3 x 2 if x < 0 if x ≥ 2 f ( x ) = { 2 x − 3 − 3 x 2 if x < 0 if x ≥ 2

Graph y = 1 x 2 y = 1 x 2 on the viewing window [ −0.5 , −0.1 ] [ −0.5 , −0.1 ] and [ 0.1 , 0.5 ] . [ 0.1 , 0.5 ] . Determine the corresponding range for the viewing window. Show the graphs.

Graph y = 1 x y = 1 x on the viewing window [ −0.5 , −0.1 ] [ −0.5 , −0.1 ] and [ 0.1 , 0.5 ] . [ 0.1 , 0.5 ] . Determine the corresponding range for the viewing window. Show the graphs.

Suppose the range of a function f f is [ −5 , 8 ] . [ −5 , 8 ] . What is the range of | f ( x ) | ? | f ( x ) | ?

Create a function in which the range is all nonnegative real numbers.

Create a function in which the domain is x > 2. x > 2.

Real-World Applications

The height h h of a projectile is a function of the time t t it is in the air. The height in feet for t t seconds is given by the function h ( t ) = −16 t 2 + 96 t . h ( t ) = −16 t 2 + 96 t . What is the domain of the function? What does the domain mean in the context of the problem?

The cost in dollars of making x x items is given by the function C ( x ) = 10 x + 500. C ( x ) = 10 x + 500.

• ⓐ The fixed cost is determined when zero items are produced. Find the fixed cost for this item.
• ⓑ What is the cost of making 25 items?
• ⓒ Suppose the maximum cost allowed is $1500. What are the domain and range of the cost function, C ( x ) ? C ( x ) ?
• 3 The Numbers: Where Data and the Movie Business Meet. “Box Office History for Horror Movies.” http://www.the-numbers.com/market/genre/Horror. Accessed 3/24/2014
• 4 http://www.eia.gov/dnav/pet/hist/LeafHandler.ashx?n=PET&s=MCRFPAK2&f=A.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites

• Authors: Jay Abramson
• Publisher/website: OpenStax
• Book title: College Algebra 2e
• Publication date: Dec 21, 2021
• Location: Houston, Texas
• Book URL: https://openstax.org/books/college-algebra-2e/pages/1-introduction-to-prerequisites
• Section URL: https://openstax.org/books/college-algebra-2e/pages/3-2-domain-and-range

© Jun 28, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

Please ensure that your password is at least 8 characters and contains each of the following:

• a special character: @$#!%*?&