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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.

Challenge

Absolute Value Functions as Piecewise Functions

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

Woman swimming in a pool
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
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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
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Next, the expressions will be simplified.
▼
Simplify
Distr

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.
Dylan's and Kriz's works
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
Distr

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
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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
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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 intercept of Since the domain of this piece domain does not contain its graph ends with an open point at
Graph of the first piece
3
Graph the Second Piece
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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
Graph of the second piece
4
Combine the Graphs on the Same Coordinate Plane
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Finally, these two pieces can be combined on the same coordinate plane.

Graph of the piecewise function
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.
A swallow drinking water while flying over a lake
External credits: sanchezn
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.
A swallow drinking water while flying over a lake
External credits: sanchezn
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.
decreasing part
For this line the intercept is and the slope is
Similarly, an equation for the increasing part can also be written.
increasing part
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

FactorOut

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.
Rio Negro Bridge
External credits: Dennis Jarvis
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
Distr

Distribute

Distribute

Add terms

Next, each function rule will be drawn separately and their graphs will be combined. First draw the graph of
Graph of the first piece
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.
Graph of the absolute value function
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
expand_more
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
expand_more
Next, the expressions will be simplified.
▼
Simplify
Distr

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