Absolute Value To Piecewise Function

Learning Outcomes

Write piecewise defined functions.
Graph piecewise-divers functions.

Sometimes, nosotros come beyond a function that requires more than than one formula in guild to obtain the given output. For example, in the toolkit functions, we introduced the absolute value function [latex]f\left(x\right)=|ten|[/latex]. With a domain of all existent numbers and a range of values greater than or equal to 0, accented value tin can be defined equally the magnitude, or modulus, of a existent number value regardless of sign. It is the distance from 0 on the number line. All of these definitions require the output to exist greater than or equal to 0.

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

[latex]f\left(10\right)=10\text{ if }ten\ge 0[/latex]

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

[latex]f\left(x\right)=-x\text{ if }x<0[/latex]

Because this requires 2 different processes or pieces, the accented value function is an instance of a piecewise function. A piecewise function is a part in which more 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 concern for which the cost per piece of a certain item is discounted one time the number ordered exceeds a certain value. Tax brackets are some other real-globe instance of piecewise functions. For example, consider a simple tax organization in which incomes up to [latex]$10,000[/latex] are taxed at [latex]ten%[/latex], and any additional income is taxed at [latex]20\%[/latex]. The tax on a total income, [latex] S[/latex] , would exist [latex]0.1S[/latex] if [latex]{S}\le$ten,000[/latex] and [latex]1000 + 0.2 (S - $ten,000)[/latex] , if [latex] S> $10,000[/latex] .

A General Notation: Piecewise Functions

A piecewise function is a function in which more than i 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 thought like this:

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

In piecewise note, the accented value function is

[latex]|10|=\begin{cases}\brainstorm{align}10&\text{ if }x\ge 0\\ -x&\text{ if }x<0\end{marshal}\end{cases}[/latex]

How To: Given a piecewise role, write the formula and place the domain for each interval.

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

Example: Writing a Piecewise Function

A museum charges $5 per person for a guided bout with a group of i to ix people or a stock-still $l fee for a group of 10 or more people. Write a office relating the number of people, [latex]north[/latex], to the cost, [latex]C[/latex].

Example: Working with a Piecewise Function

A cell phone visitor uses the function beneath to determine the price, [latex]C[/latex], in dollars for [latex]k[/latex] gigabytes of data transfer.

[latex]C\left(thousand\right)=\begin{cases}\begin{align}{25} \hspace{2mm}&\text{ if }\hspace{2mm}{ 0 }<{ g }<{ two }\\ { 25+10 }\left(g - 2\right) \hspace{2mm}&\text{ if }\hspace{2mm}{ g}\ge{ 2 }\finish{marshal}\end{cases}[/latex]

Find the toll of using ane.5 gigabytes of data and the toll of using four gigabytes of data.

How To: Given a piecewise part, sketch a graph.

Betoken on the [latex]ten[/latex]-axis the boundaries defined past 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 slice. Exercise not graph two functions over i interval because it would violate the criteria of a function.

Example: Graphing a Piecewise Function

Sketch a graph of the function.

[latex]f\left(x\correct)=\begin{cases}\begin{align}{ x }^{ii} \hspace{2mm}&\text{ if }\hspace{2mm}{ ten }\le{ ane }\\ { 3 } \hspace{2mm}&\text{ if }\hspace{2mm} { 1 }<{ x }\le 2\\ { x } \hspace{2mm}&\text{ if }\hspace{2mm}{ x }>{ two }\stop{align}\cease{cases}[/latex]

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Graph the post-obit piecewise function.

[latex]f\left(x\correct)=\begin{cases}\begin{align}{ x}^{3} \hspace{2mm}&\text{ if }\hspace{2mm}{ x }<{-one }\\ { -2 } \hspace{2mm}&\text{ if } \hspace{2mm}{ -1 }<{ x }<{ 4 }\\ \sqrt{x} \hspace{2mm}&\text{ if }\hspace{2mm}{ x }>{ 4 }\finish{align}\end{cases}[/latex]

Try It

You lot can utilise an online graphing tool to graph piecewise defined functions. Picket this tutorial video to learn how.

Graph the following piecewise function with an online graphing tool.

[latex]f\left(x\right)=\begin{cases}\brainstorm{marshal}{ x}^{iii} \hspace{2mm}&\text{ if }\hspace{2mm}{ x }<{-ane }\\ { -2 } \hspace{2mm}&\text{ if } \hspace{2mm}{ -1 }<{ x }<{ 4 }\\ \sqrt{10} \hspace{2mm}&\text{ if }\hspace{2mm}{ x }>{ 4 }\end{align}\cease{cases}[/latex]

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Q&A

Tin more than 1 formula from a piecewise role be practical to a value in the domain?

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

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