graphing piecewise functions homework

Chapter 2: Functions and Their Graphs

Section 2.4: library of functions; piecewise functions, learning outcomes.

• Identify base functions
• Graph piecewise-defined functions.

Identifying Base Functions

In this text we will be exploring functions—the shapes of their graphs, their unique characteristics, their algebraic formulas, and how to solve problems with them. When learning to read, we start with the alphabet. When learning to do arithmetic, we start with numbers. When working with functions, it is similarly helpful to have a library of building-block elements. We call these our “base functions,” which form a set of basic named functions for which we know the graph, formula, and special properties. Some of these functions are programmed to individual buttons on many calculators. For these definitions we will use [latex]x[/latex] as the input variable and [latex]y=f\left(x\right)[/latex] as the output variable.

We will see these base functions, combinations of base functions, their graphs, and their transformations frequently throughout this book. It will be very helpful if we can recognize these toolkit functions and their features quickly by name, formula, graph, and basic table properties. The graphs and sample table values are included with each function shown below.

Key Equations

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 [latex]f\left(x\right)=|x|[/latex]. 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, would be 0.1S if [latex]{S}\le\[/latex] $10,000 and 1000 + 0.2 (S – $10,000), if S> $10,000.

A General Note: 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:

[latex] f\left(x\right)=\begin{cases}\text{formula 1 if x is in domain 1}\\ \text{formula 2 if x is in domain 2}\\ \text{formula 3 if x is in domain 3}\end{cases} [/latex]

In piecewise notation, the absolute value function is

[latex]|x|=\begin{cases}\begin{align}&x&&\text{ if }x\ge 0\\ &-x&&\text{ if }x<0\end{align}\end{cases}[/latex]

How To: 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.

Example 1: 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, [latex]n[/latex], to the cost, [latex]C[/latex].

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

[latex]C(n)=\begin{cases}\begin{align}{5n}&\hspace{5mm}\text{ if }{0}<{n}<{10}\\ 50&\hspace{5mm}\text{ if }{n}\ge 10\end{align}\end{cases}[/latex]

Analysis of the Solution

The function is represented in Figure 1. The graph is a diagonal line from [latex]n=0[/latex] to [latex]n=10[/latex] and a constant after that. In this example, the two formulas agree at the meeting point where [latex]n=10[/latex], but not all piecewise functions have this property.

Example 2: Working with a Piecewise Function

A cell phone company uses the function below to determine the cost, [latex]C[/latex], in dollars for [latex]g[/latex] 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), 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.

[latex]C(1.5) = \$25[/latex]

To find the cost of using 4 gigabytes of data, 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 2. We can see where the function changes from a constant to a shifted and stretched identity at [latex]g=2[/latex]. We plot the graphs for the different formulas on a common set of axes, making sure each formula is applied on its proper domain.

How To: 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.

Example 3: Graphing a Piecewise Function

Sketch a graph of the function.

[latex]f\left(x\right)=\begin{cases}\begin{align}&{ x }^{2} &&\hspace{-5mm}\text{ if }{ x }\le{ 1 }\\ &{ 3 } &&\hspace{-5mm}\text{ if } { 1 }<{ x }\le 2\\ &{ x } &&\hspace{-5mm}\text{ if }{ x }>{ 2 }\end{align}\end{cases}[/latex]

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.

Below are 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.

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

Graph the following piecewise function.

[latex]f\left(x\right)=\begin{cases}{ x}^{3} \text{ if }{ x }<{-1 }\\ { -2 } \text{ if } { -1 }<{ x }<{ 4 }\\ \sqrt{x} \text{ if }{ x }>{ 4 }\end{cases}[/latex]

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.

Key Concepts

• A piecewise function is described by more than one formula.
• A piecewise function can be graphed using each algebraic formula on its assigned subdomain.

Section 2.4 Homework Exercises

1. How do you graph a piecewise function?

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

2. [latex]f(x)=\begin{cases}{x}+{1}&\text{ if }&{ x }<{ -2 } \\{-2x - 3}&\text{ if }&{ x }\ge { -2 }\\ \end{cases} [/latex]

3. [latex]f\left(x\right)=\begin{cases}{2x - 1}&\text{ if }&{ x }<{ 1 }\\ {1+x }&\text{ if }&{ x }\ge{ 1 } \end{cases}[/latex]

4. [latex]f\left(x\right)=\begin{cases}{x+1}&\text{ if }&{ x }<{ 0 }\\ {x - 1 }&\text{ if }&{ x }>{ 0 }\end{cases}[/latex]

5. [latex]f\left(x\right)=\begin{cases}{3} &\text{ if }&{ x } <{ 0 }\\ \sqrt{x}&\text{ if }&{ x }\ge { 0 }\end{cases}[/latex]

6. [latex]f\left(x\right)=\begin{cases}{x}^{2}&\text{ if }&{ x } <{ 0 }\\ {1-x}&\text{ if }&{ x } >{ 0 }\end{cases}[/latex]

7. [latex]f\left(x\right)=\begin{cases}{x}^{2}&\text{ if }&{ x }<{ 0 }\\ {x+2 }&\text{ if }&{ x }\ge { 0 }\end{cases}[/latex]

8. [latex]f\left(x\right)=\begin{cases}x+1& \text{if}& x<1\\ {x}^{3}& \text{if}& x\ge 1\end{cases}[/latex]

9. [latex]f\left(x\right)=\begin{cases}|x|&\text{ if }&{ x }<{ 2 }\\ { 1 }&\text{ if }&{ x }\ge{ 2 }\end{cases}[/latex]

For the following exercises, given each function [latex]f[/latex], evaluate [latex]f\left(-3\right),f\left(-2\right),f\left(-1\right)[/latex], and [latex]f\left(0\right)[/latex].

10. [latex]f\left(x\right)=\begin{cases}{ x+1 }&\text{ if }&{ x }<{ -2 }\\ { -2x - 3 }&\text{ if }&{ x }\ge{ -2 }\end{cases}[/latex]

11. [latex]f\left(x\right)=\begin{cases}{ 1 }&\text{ if }&{ x }\le{ -3 }\\{ 0 }&\text{ if }&{ x }>{ -3 }\end{cases}[/latex]

12. [latex]f\left(x\right)=\begin{cases}{-2}{x}^{2}+{ 3 }&\text{ if }&{ x }\le { -1 }\\ { 5x } - { 7 } &\text{ if }&{ x } > { -1 }\end{cases}[/latex]

For the following exercises, given each function [latex]f[/latex], evaluate [latex]f\left(-1\right),f\left(0\right),f\left(2\right)[/latex], and [latex]f\left(4\right)[/latex].

13. [latex]f\left(x\right)=\begin{cases}{ 7x+3 }&\text{ if }&{ x }<{ 0 }\\{ 7x+6 }&\text{ if }&{ x }\ge{ 0 }\end{cases}[/latex]

14. [latex]f\left(x\right)=\begin{cases}{x}^{2}{ -2 }&\text{ if }&{ x }<{ 2 }\\{ 4+|x - 5|}&\text{ if }&{ x }\ge{ 2 }\end{cases}[/latex]

15. [latex]f\left(x\right)=\begin{cases}5x& \text{if}& x<0\\ 3& \text{if}& 0\le x\le 3\\ {x}^{2}& \text{if}& x>3\end{cases}[/latex]

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

16. [latex]f\left(x\right)=\begin{cases}{x+1}&\text{ if }&{ x }<{ -2 }\\{ -2x - 3}&\text{ if }&{ x }\ge{ -2 }\end{cases}[/latex]

17. [latex]f\left(x\right)=\begin{cases}{x}^{2}{ -2 }&\text{ if}&{ x }<{ 1 }\\{-x}^{2}+{2}&\text{ if }&{ x }>{ 1 }\end{cases}[/latex]

18. [latex]f\left(x\right)=\begin{cases}{ 2x - 3 }&\text{ if }&{ x }<{ 0 }\\{ -3}{x}^{2}&\text{ if }&{ x }\ge{ 2 }\end{cases}[/latex]

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Graphing Piecewise and Step Functions

Abilities covered in this lesson, lesson settings, catch-up and review.

Here are a few recommended readings before getting started with this lesson.

• Domain of a Function
• Linear Functions
• Linear Inequalities
• Graphing Linear Functions
• Graphing Linear Inequalities

How Many Cans Are There in the Fridge?

Mark always likes to have cans of his favorite soft drink in the refrigerator so that he can enjoy a cold drink whenever he wants.

At the beginning of the week, he had 11 cans in the fridge. Over the week, Mark wrote down each time cans were consumed or added.

• On Monday, Mark drank 1 can.
• On Wednesday, Mark bought 2 cans at a convenience store and put them into the fridge.
• On Friday, Mark had a party and a total of 10 cans were consumed.
• On Saturday, Mark bought a 12-pack of soda from a supermarket and put them away in the refrigerator.

Piecewise Functions

A piecewise function is a function that is defined differently for different parts of its domain . These functions are commonly defined using intervals or inequalities . Consider the following example. f(x) = - x-2, & if x < 0 2x+1, & if x≥ 0 This function is defined as one linear function for values of x less than 0, and a different linear function for values of x that are at least zero. The graph of f is obtained by graphing the two rays . It should be noted that both rays represent the same function.

Practice Finding Values

A piecewise function f is shown below. For every given value of x, find the correct value of f(x). Round the result to two decimal places , if necessary.

Graphing Piecewise Functions

When graphing a piecewise function , each piece must be considered separately. First, a piece is graphed for the values of its domain . Then an end — or ends — of the piece are marked with one of the following.

• A closed circle for an x-value for which the function is defined.
• An open circle for an x-value for which the function is undefined .

The first piece is f(x) = 4x + 6. This is a linear function written in slope-intercept form . The function can be graphed by using the y-intercept and the slope .

Now, since the function is limited to inputs less than or equal to -1, the line will be graphed until it reaches x = -1. Since the inequality x≤ -1 is non-strict , the function is defined for x=-1 and the circle will be closed.

A similar process can be repeated to graph the second piece f(x) =- x+3. First, the line will be graphed.

This piece is defined for values of x greater than -1. This means that the line will be drawn starting at x=-1. Also, since the inequality is strict , the circle will be open.

Finally, the pieces will be graphed together in the same coordinate plane to complete the graph. It is important to pay attention to the limit points, since each value of x must be assigned to only one value of the function.

Going to the Beach

f(t) = 15t,& if 0≤ t ≤ 1 ? & ? & Since Heichi was biking at 15 miles per hour for one hour, he traveled 15 miles in this interval. In the next piece, the value of the function is constant at 15 for the values of t between 1 and 6 because he is just hanging out at the beach for these 5 hours. f(t) = 15t,& if 0≤ t ≤ 1 15, & if 1 < t ≤ 6 ? & Finally, when Heichi goes home, he bikes at 10 miles per hour. Since he is returning home, the distance from his house will decrease. Therefore, the distance traveled is written as the product of -10 and t. -10 t Also, since Heichi arrives at home one and a half hours later, the final value of the last piece should be 7.5 hours after the beginning of the trip. Therefore, the function needs to be translated 7.5 units right by subtracting 7.5 from t before multiplying by -10. - 10(t-7.5) = -10t + 75 Finally, it is possible to write the complete function rule by adding this final piece. f(t) = 15t,& if 0≤ t ≤ 1 15, & if 1 < t ≤ 6 -10t+75, & if6 < t ≤ 7.5

The second piece of the function is the horizontal line f(t) = 15. Also, since the inequality is non-strict, the circle at t=6 will be filled.

Now, the x-intercept can be used to graph the third and final piece. From Part A, it is known that the x-intercept is 7.5.

Finally, the graph of the piecewise function is completed.

Domain [0,7.5] The function's output represents how far Heichi is from his house. Looking at the graph, it can be seen that the minimum value of the function is 0, when Heichi is at his house, and the maximum value is 15, when Heichi is at the beach. While he is biking between his house and the beach, all distances from 0 to 15 are reached.

Step Functions

A step function is a piecewise function that is defined by a constant value on each part of its domain . As an example, consider the following function . f(x) = 0, & if 0≤ x < 1 2, & if 1≤ x < 2 4, & if 2≤ x < 3 6, & if 3≤ x < 4 The graph of a step function consists of horizontal line segments , which can be interpreted as steps. The graph of the given function has four line segments.

Collecting Boxes Over the Week

As a summer activity, Tearrik participates in charity events for his community. He is volunteering for a food drive event this weekend.

He went to a shopping center multiple times over the week to collect boxes of food for the food drive. On each day, he collected the following number of boxes.

• 5 boxes on Monday
• 3 boxes on Tuesday
• 2 boxes on Wednesday
• 3 boxes on Thursday
• 4 boxes on Friday

f(x) = 0, & if 0≤ x < 1 5, & if 1≤ x < 2 8, & if 2≤ x < 3 10, & if 3≤ x < 4 13, & if 4≤ x < 5 17, & if 5≤ x < 6

Use the given days of the week as the domain of the function .

First, the domain of the function must be determined. The domain can be defined as the given days of the week, starting with the Sunday before Tearrik started collecting the boxes as x=0. Since Tearrik makes his first pickup on Monday, he starts with zero boxes on Sunday x=0. f(x) = 0, if 0≤ x < 1 On Monday, since one day has passed from Sunday, x=1. On this day, Tearrik collected his first 5 boxes. f(x) = 5, if 1≤ x < 2 On the following days, Tearrik collected more boxes. The boxes picked up each day were added to the number of boxes collected previously. Using this information, the pieces can be written in a table .

Finally, the function rule of the step function can be written based on the table. f(x) = 0, & if 0≤ x < 1 5, & if 1≤ x < 2 8, & if 2≤ x < 3 10, & if 3≤ x < 4 13, & if 4≤ x < 5 17, & if 5≤ x < 6

Graphing Step Functions

Since step functions are piecewise functions , to graph them, each piece must be considered separately. First, each piece is graphed in its domain as a part of a horizontal line . The end or ends are then marked with one of the following.

• A closed circle for an endpoint that is included in the function rule .
• An open circle for an endpoint that is not included in the function rule.

The first piece is defined over the interval 0≤ x < 2. This piece will be graphed by drawing a horizontal line at y=1 from x=0 to x=2.

It can be noted that x=0 is included in the domain of the first piece, but x=2 is not. Therefore, the left end of the segment will be marked with a closed circle and the right end with an open circle.

Then, the same process is repeated for each piece of the function.

Saving Tips for a Week

Zain is working as a server in a restaurant for a week before their summer vacation ends.

Most of their payment comes from the tips they receive. Zain made note of how much they received in tips, starting from Monday and through their last day working on Saturday.

• $49.50 on Monday
• $38.17 on Tuesday
• $41.45 on Wednesday
• $58.33 on Thursday
• $60.55 on Friday
• $57.00 on Saturday

f(x) = 0, if0 ≤ x < 1 This indicates that Zain received $0 in tips on Sunday before they start working. On Monday x=1, Zain received $49.50 in tips. This defines the next piece of the function. f(x) = 49.50, if1≤ x < 2 Since the function reflects the total amount of money Zain receives in tips, the value of each of the pieces can be determined by adding the new tip value to the total tip amount from the day before. Using this information, the pieces can be written in a table .

Finally, the step function can be written based on the table. f(x) = 0, &if 0≤ x < 1 49.50, &if 1 ≤ x < 2 87.67, &if 2 ≤ x < 3 129.12, &if 3 ≤ x < 4 187.45, &if 4 ≤ x < 5 248.00, &if 5 ≤ x < 6 305.00, &if 6 ≤ x < 7

The next piece is the horizontal segment at 49.50 with a closed endpoint at x=1 and an open endpoint at x=2.

The remaining pieces can be added to the graph by following the same process.

Greatest Integer Function

The greatest integer function , also known as the floor function , assigns the largest integer that is less than or equal to the value of x. This function is usually written as f(x)= ⌊ x ⌋ or f(x)= ||x||. Consider the following examples. ⌊ - 2 ⌋ & = - 2 ⌊ - 1.5 ⌋ & = - 2 ⌊ -0.01 ⌋ & = -1 ⌊ 1 ⌋ & = 1 ⌊ 1.25 ⌋ & = 1 It can be noted that if x is an integer, the function returns the same value. ⌊ - 2⌋ = - 2 ⌊ 0⌋ = 0 ⌊ 1 ⌋ = 1 Otherwise, it returns the closest integer at the left of x in a number line .

Considering more values can help understand how to draw the graph of the function.

From the table above, it can be seen that the function only changes its value when a new integer is reached. It can be noted that the greatest integer function is a step function . Its graph is presented as follows.

The Price of Parking at the Cinema

Dominika is going to a movie at a local theater on her last day of vacation.

The cost to park in the theater lot is $10 for less than an hour. An additional $2.50 is charged for each hour of parking.

f(x) = 2.5 ⌊ x ⌋ + 10

f_1(x) = 2.5 ⌊ x ⌋ Additionally, the lot requires an initial payment of $10 for the first 59 minutes of parking. This information can be used to add 10 to the value of the obtained function f_1. f(x) = 2.5 ⌊ x ⌋ + 10 This completes the required function, as the price to enter is considered and $2.50 is added as each hour is reached.

Next, since the greatest integer function is multiplied by 2.5, each value of y is multiplied by 2.5, vertically stretching the spaces between each horizontal segment.

Then, since 10 is added to the product, each segment is translated vertically 10 units up.

Finally, the scope of the coordinate plane will be adjusted so that more steps of the graph can be seen.

Math Graphing Assignment

During his summer vacation, Ignacio went to private math lessons. After learning about the greatest integer function , Ignacio was asked by his math tutor to graph the numbers y greater than or equal to ⌊ x ⌋.

Graph the greatest integer function. Then, determine the solution set by shading the appropriate region.

The first step to graph an inequality is to graph the border function . The border function of the given inequality is given by the greatest integer function. This function is a step function whose output is the greatest integer less than or equal to the input x. Note that the inequality is non-strict , so the horizontal lines are drawn as solid lines.

x= 1, y= 0.5

Finally, the following is the complete graph of the inequality that the teacher asked for.

Cans in the Fridge

At the beginning of this lesson, it was asked that a function for the number of soda cans in Mark's refrigerator be written. Mark starts the week with 11 cans in the fridge and then does the following.

f(x) = 11, if 0 ≤ x < 1 Mark drank one can of soda on Monday. Therefore, he has now 11 - 1 = 10 cans left in the fridge that day. Also, since he neither drank nor bought any soda on Tuesday, the function's output is still 10 until Wednesday, when x=3. f(x) = 10, if1 ≤ x < 3 Mark bought 2 cans on Wednesday, meaning that there are 10 + 2 =12 cans of soda in the fridge that day. Since he neither bought nor drank any of the soda in the refrigerator on Thursday, he has 12 cans until Friday. f(x) = 12, if 3 ≤ x < 5 At Friday's party, a total of 10 cans were drunk. Therefore, he has 12 - 10 = 2 cans left in the refrigerator for Saturday. Then, on Saturday, he bought a 12-pack of soda cans, making it for 2 + 12 = 14 cans in the refrigerator for that day. Because no more information was given, the whole function rule can now be written. f(x) = 11, & if 0 ≤ x < 1 10, & if 1 ≤ x < 3 12, & if 3 ≤ x < 5 2, & if 5 ≤ x < 6 14, & if 6 ≤ x < 7

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