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Some functions may have different rules for different parts of their domain. In this lesson, it will be discussed whether an absolute value function can be written as a piecewise function or not.

Catch-Up and Review

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

Graphing linear functions in slope-intercept form
Writing the equation of a line in slope-intercept form from a graph
Absolute Value Function
Absolute Value Inequalities in Two Variables
Piecewise Function
Graphing Piecewise Functions

Challenge

Absolute Value Functions as Piecewise Functions

Jordan is getting ready for the inter-class swimming competition at her school.

She swims to the far end of the pool and comes back to the starting point. The function below models Jordan's distance from the far end of the pool after

seconds.

a Rewrite the given absolute value function as a piecewise function.

b Draw the graph of the absolute value function as a piecewise function and state a reasonable domain and range for the context.

Explore

Graphs of Piecewise Functions

Examine the given piecewise functions and inequalities on the left. Match them with their corresponding graph.

Is it possible to represent these graphs with absolute value functions or absolute value inequalities? If so, write these functions.

Discussion

Writing an Absolute Value Function as a Piecewise Function

An expression involving an absolute value can be defined as follows.

Using this definition, an absolute value function can be written as a piecewise function. Consider an example function.

The above function will be rewritten as a piecewise function. The procedure can be completed in two steps.

1

Use the Definition of Absolute Value

First, identify the expression that involves absolute value.

This expression is equivalent to

if

is less than

Conversely, this expression is equivalent to

if

is greater than or equal to

Now this piecewise definition of the absolute value expression can be used for the given function.

2

Simplify Expressions and Inequalities

Next, the expressions will be simplified.

▼

Simplify

Distribute

Distribute

Add terms

Finally, the inequalities describing the domains of the pieces will be rearranged. To do so, subtract

from both sides of the inequalities, then multiply both sides by

The given absolute value function has been written as a piecewise function.

Absolute Value Function	Piecewise Function

Example

Who Correctly Rewrote an Absolute Value Function as a Piecewise Function?

Dylan and Kriz have been asked to write the following absolute value function as a piecewise function.

The functions they wrote are shown in the diagram. Who correctly wrote the given function as a piecewise function?

Hint

Start by identifying the absolute value expression.

Solution

First, the given function will be written as a piecewise function. Then it can be seen who is correct. Consider the absolute value function.

By using the definition of an absolute value the expression,

can be rewritten. If

is less than

its absolute value is equal to its opposite value. Conversely, if

is greater than or equal to

its absolute value is equal to itself.

With this information in mind, the absolute value function can be rewritten.

Next, the expressions need to be simplified.

▼

Simplify

Distribute

Distribute

Add terms

Finally, the domain of this piecewise function should be rearranged. First,

will be subtracted from both sides of the inequalities.

By dividing the inequalities by

the parts of the domain can be identified. Recall that dividing an inequality by a negative number reverses the inequality symbol.

The absolute value function is now completely rewritten as a piecewise function.

Absolute Value Function	Piecewise Function

Comparing this function with the functions written by Dylan and Kriz, it appears that Kriz wrote it correctly.

Discussion

Graphing an Absolute Value Function as a Piecewise Function

Absolute value functions can be written as piecewise functions. By graphing the pieces for their domains, the graph of the absolute value function can be obtained. As an example, the following function will be graphed.

Its graph can be drawn in four steps.

1

Write the Absolute Value Function as a Piecewise Function

By the definition of absolute value, an absolute value expression can be divided into two. With this in mind, two function rules for the given absolute value function can be defined.

After simplifying the expressions and inequalities, a piecewise-defined function is obtained.

To graph the function, each individual piece will be graphed and then combined on the same coordinate plane.

2

Graph the First Piece

First the graph of

will be drawn for the domain

Notice that this piece is written in slope-intercept form.

This function has a slope of

and a

Since the domain of this piece domain does not contain

its graph ends with an open point at

3

Graph the Second Piece

Now, the other piece will be drawn for the domain

Using its slope

and

intercept

the graph of the second piece can be drawn. Since this time

belongs to the domain, the graph of this piece ends with a closed point at

4

Combine the Graphs on the Same Coordinate Plane

Finally, these two pieces can be combined on the same coordinate plane.

Example

Modeling the Wings of a Swallow Using an Absolute Value Function

LaShay likes to make connections between the shapes she finds in daily life and the concepts she encounters in her math lessons. While watching a documentary about swallows, LaShay thinks that the wings of a swallow can be modeled by an absolute value function.

Error loading image.

The axes represent lengths in inches.

a If the tip of one wing is at

and the swallow's head is at

write a piecewise function that models the wings.

b Rewrite the function as an absolute value function and state its domain.

Answer

a Piecewise Function:

b Absolute Value Function:

Domain:

Hint

a An absolute value function is symmetric across the vertical line passing through its vertex. Use this symmetry to find another point on the graph.

b Rearrange the rule for each piece so that one rule contains an expression and the other rule contains its opposite.

Solution

a The points

and

are on the graph of the absolute value function. Since an absolute value function is symmetric across the vertical line passing through its vertex, one more point on the graph can be found.

Error loading image.

The point

is on the graph. Using these three points, two function rules can be written. One for the decreasing part, and the other for the increasing part of the graph. The domain of these parts can be written as follows.

Note the point at

can belong to either piece, so long as it belongs to only one of them. The next step will then be to find the equation for both lines. For this, recall the slope-intercept form of a linear function.

In this form

is the slope of the line and

is the

intercept. Start with the decreasing part. The

intercept and slope can be found from the graph. For this line the

intercept is

and the slope is

Similarly, an equation for the increasing part can also be written. For the increasing line, the

intercept is

and the slope is

Knowing the equations for both lines, the absolute value function can be written as a piecewise function.

b In this part the piecewise function will be used to write an absolute value function.

To do so, the function rules can be rearranged so that one rule contains an expression and the other rule contains its opposite.

Factor out

Factor out

The function rules contain the expressions

and

These expressions produce non-negative values in their domains. Therefore, they can be written using absolute values.

The domain of this absolute value function is the union of the domains for the function rules.

Example

Modeling Rio Negro Bridge Using an Absolute Value Function

The Rio Negro Bridge is a meter long cable-stayed bridge over the Rio Negro in Brazil.

Error loading image.

Kevin researches the bridge on the internet and writes an absolute value function for the path of the two longest cables.

Here,

is the horizontal distance to the leftmost point of the path and

is the height of the path.

a Write the given absolute value function as a piecewise function and graph it.

b If all measures are in meters, what is the distance between the leftmost and the rightmost points of the path?

Answer

a Function:

Graph:

b

meters

Hint

a Use the definition of absolute values to write a piecewise function.

b Identify the points whose

coordinates are zero.

Solution

a To rewrite the given function, the definition of absolute value will be used.

The function rules can be simplified.

▼

Simplify

Distribute

Distribute

Add terms

Next, each function rule will be drawn separately and their graphs will be combined. First draw the graph of

The graph ends with an open circle because its domain is the set of

values less than

For

values greater than or equal to

the graph of

will be drawn. This is a linear function written in slope-intercept form.

Using this information, its graph can be drawn.

This piece ends with a closed circle, as

is in its domain. Finally, both graphs will be combined on the same coordinate plane.

b The leftmost and the rightmost point of the path are

and

Since all the measures are in meters, the distance between these two points is meters.

Discussion

Writing an Absolute Value Inequality as a Piecewise Inequality

An absolute value inequality can be formed by replacing the equals sign in an absolute value function with an inequality symbol. Therefore, writing an absolute value inequality as a piecewise inequality can be compared to writing an absolute value function as a piecewise function. Consider an example absolute value inequality.

This inequality can be rewritten as a piecewise inequality in two steps.

1

Use the Definition of Absolute Value

First, identify the expression that involves the absolute value.

If

is greater than or equal to

then its absolute value is equal to itself. Conversely, if

is less than

then its absolute value is equal to its opposite value.

Now, the piecewise definition of the absolute value expression can be used for the given inequality.

2

Simplify Expressions and Inequalities

Next, the expressions will be simplified.

▼

Simplify

Distribute

Distribute

Subtract terms

Finally, the inequalities used to describe the parts of the domain will be rearranged. To do so, add

to both sides and then divide both sides by

The given absolute value inequality has been written as a piecewise inequality.

Discussion

Graphing an Absolute Value Inequality as a Piecewise Inequality

Absolute value inequalities in two variables can be written as piecewise inequalities. By drawing the graph of each piece in the piecewise inequality, the graph of the absolute value inequality is also drawn. Consider an absolute value inequality.

This inequality can be graphed as a piecewise inequality in four steps.

1

Write the Absolute Inequality as a Piecewise Inequality

Use the definition of absolute values to split the inequality into two pieces.

Next, the expressions and the inequalities will be simplified.

▼

Simplify

Distribute

Distribute

Add terms

Now, the inequalities describing the domain of the pieces will be rearranged. To do so, subtract

from both sides of the inequalities.

To graph the piecewise-defined inequality, first each individual piece of the inequality will be drawn. Then the graphs will be combined on the same coordinate plane.

2

Graph the First Piece

Begin by determining the boundary line. This line can be found by replacing the inequality symbol with the equals sign.

The boundary line is already written in slope-intercept form. Using the slope and

intercept, the line can be graphed. Since the inequality is strict, the line will be dashed. The domain for the boundary line contains

values less than

This means that the only part of the graph that should be considered is to the left of

Additionally, the boundary line will have an open endpoint at

Next, the region to be shaded will be determined. To do so, choose a point whose

coordinate is less than

but is not on the boundary line — for example,

,

▼

Evaluate

Add terms

\(0 < 1\ {\color{#009600}{\bm{\Large{\checkmark }}}}\)

Since the point satisfies the inequality, the region that contains the point will be shaded.

3

Graph the Second Piece

The second piece can be graphed similarly. First, write the equation of the boundary line.

The slope and the

intercept of the boundary line are

and

respectively. Since the inequality is strict, the boundary line will be dashed. The domain for the boundary line contains the

values greater than or equal to

This means that the only part of the graph that should be considered is to the right of

Additionally, the boundary line will have a closed endpoint at

Now, to determine which region must be shaded choose a point with a

coordinate greater than

but that is not on the line. For example,

can be used.

,

▼

Evaluate

Zero Property of Multiplication

Subtract term

\(0 \nless \text{-} 1\ {\color{#FF0000}{\bm{\Large{\times }}}}\)

Since the point does not satisfy the inequality, the region that does not contain the point will be shaded.

4

Combine the Graphs on the Same Coordinate Plane

Finally, the graphs of the two pieces can be combined on the same coordinate plane.

Example

Absolute Value Inequality as Piecewise Inequality

Maya notices that the region illuminated by a car's left headlight can be modeled by an absolute value inequality.

Error loading image.

The inequality below models this region.

a If the right headlight is

unit away from the left, write a piecewise-defined inequality for the right headlight.

b Graph the piecewise-defined inequality for the right headlight.

Answer

a

b

Hint

a Start by writing an absolute value inequality to represent the right headlight. This is a horizontal translation of the given absolute value inequality.

b To graph the absolute value inequality as a piecewise inequality, start by drawing each piece for its domain and then combine the pieces on the same coordinate plane.

Solution

a An inequality for the right headlight can be obtained by a horizontal translation of the given absolute value inequality. Since the right headlight is

unit to the right of the left headlight, the given inequality needs to be translated

unit to the right. This can be done by subtracting

from the input of the rule.

Considering the definition of absolute values, this inequality can be divided into two pieces.

The expressions can be simplified.

▼

Simplify

Distribute

Distribute

Subtract terms

Now, the inequalities describing the domains of the pieces will be rearranged. To do so,

will be added to both sides of the inequalities.

b Consider the first piece of the inequality. The domain for this piece and the boundary line can be written.

The boundary line is in slope-intercept form. Its graph can be drawn by using the slope and

intercept. Since the inequality is not strict, the boundary line will be solid. Additionally, since

is less than

the boundary line will have an open endpoint at

To determine the region to be shaded, choose a test point and substitute it into the inequality — for example,

,

▼

Evaluate right-hand side

Zero Property of Multiplication

Identity Property of Addition

\(0 \ngeq 6\ {\color{#FF0000}{\bm{\Large{\times }}}}\)

Since the point does not satisfy the inequality, the region that does not contain the test point will be shaded. Note that only points in the corresponding domain

will be considered. Next, the second piece of the inequality will be graphed.

The boundary line's slope is

Also,

satisfies the second piece. This point is the leftmost point of the boundary line because its domain is

Additionally, since the inequality is non-strict, the boundary line will be solid. Now, choose a point not on the boundary line whose

coordinate is greater than or equal to

— for example,

,

▼

Evaluate right-hand side

Multiply

Subtract term

\(0 \ngeq 11\ {\color{#FF0000}{\bm{\Large{\times }}}}\)

Since the point did not satisfy the inequality, the region that does not contain the point that will be shaded. Finally, the graphs of the two pieces can be combined on the same coordinate plane.

Maya can now complete her image.

Error loading image.

Closure

Absolute Value Functions as Piecewise Functions

Considering the methods and examples discussed in this lesson, the challenge presented at the start can now be solved. Jordan swims to the far end of the pool and comes back to the starting point. The absolute value function that models Jordan's distance from the far end after seconds is given.

a Rewrite the given function as a piecewise function.

b Draw the graph of the absolute value function as a piecewise function and state a reasonable domain and range for the function.

Answer

a

b Graph:

Domain:
Range:

Hint

a Use the definition of absolute value to write the given function as a piecewise function.

b Determine the slope of each piece. What is the length of the pool? Use it to write a range.

Solution

a Consider the absolute value function that will be rewritten as a piecewise function.

The absolute value expression in this function can be divided into two cases.

If is less than then its absolute value is equal to its opposite value.
If is greater than or equal to then its absolute value is equal to itself.

With this information in mind, the absolute value function can be written as a piecewise function.

Next, the function rules need to be simplified.

▼

Simplify

Distribute

Distribute

The inequalities can be solved for

To do so,

will be added to both sides of the inequalities.

b The piecewise function will now be graphed. To do so, each individual piece will be graphed and then combined on one coordinate plane.

First graph

This function has a slope of

and a

Since the domain for this piece is the set of numbers less than

the graph should end with an open point at

Similarly, the other piece can be drawn. Its slope is The domain of this piece contains the values greater than or equal to Therefore, it has a closed endpoint at

The combination of the above graphs is the graph of the piecewise function.

Since

represents time and

represents distance, they cannot be negative. Therefore, both

and

are non-negative numbers.

Recall that Jordan swims to the far end of the pool and comes back to the starting point. At

Jordan is at the starting point. Therefore, by substituting

into the given function the length of the pool can be found.

▼

Evaluate right-hand side

Subtract term

Multiply

At

Jordan is

meters away from the end. With this information it can be said that

When Jordan is back to the starting point, she is again

meters away from the far end. Therefore, to calculate how many seconds it takes her to swim to the far end and come back to the starting point,

will be substituted in the given function.

▼

Solve for

Rearrange equation

State solutions

Jordan is

meters away from the far end at

and

Therefore, it takes her

seconds to swim to the far end and come back to the starting point. With this information it can be said that

so the domain and the range can be written.

This means that only the first quadrant is relevant.

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