# 3.2 Domain and range  (Page 6/11)

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## Finding the domain and range using toolkit functions

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=2{x}^{3}-x.$

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

The domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right)\text{\hspace{0.17em}}$ and the range is also $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).$

## Finding the domain and range

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=\frac{2}{x+1}.$

We cannot evaluate the function at $\text{\hspace{0.17em}}-1\text{\hspace{0.17em}}$ because division by zero is undefined. The domain is $\text{\hspace{0.17em}}\left(-\infty ,-1\right)\cup \left(-1,\infty \right).\text{\hspace{0.17em}}$ Because the function is never zero, we exclude 0 from the range. The range is $\text{\hspace{0.17em}}\left(-\infty ,0\right)\cup \left(0,\infty \right).$

## Finding the domain and range

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=2\sqrt{x+4}.$

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

The domain of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left[-4,\infty \right).$

We then find the range. We know that $\text{\hspace{0.17em}}f\left(-4\right)=0,\text{\hspace{0.17em}}$ and the function value increases as $\text{\hspace{0.17em}}x\text{\hspace{0.17em}}$ increases without any upper limit. We conclude that the range of $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ is $\text{\hspace{0.17em}}\left[0,\infty \right).$

Find the domain and range of $\text{\hspace{0.17em}}f\left(x\right)=-\sqrt{2-x}.$

domain: $\text{\hspace{0.17em}}\left(-\infty ,2\right];\text{\hspace{0.17em}}$ range: $\text{\hspace{0.17em}}\left(-\infty ,0\right]$

## 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 $\text{\hspace{0.17em}}f\left(x\right)=|x|.\text{\hspace{0.17em}}$ 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.

$f\left(x\right)=x\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x\ge 0$

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

$f\left(x\right)=-x\text{\hspace{0.17em}}\text{if}\text{\hspace{0.17em}}x<0$

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 $\text{\hspace{0.17em}}S\text{\hspace{0.17em}}$ would be $\text{\hspace{0.17em}}0.1S\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}S\le \text{}10\text{,}000\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}\text{}1000+0.2\left(S-\text{}10\text{,}000\right)\text{\hspace{0.17em}}$ if $\text{\hspace{0.17em}}S>\text{}10\text{,}000.$

## 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.

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

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