# Piecewise-Defined Functions

A piecewise function exists when a function is defined by two or more different functions throughout its domain. The first step in evaluating a piecewise function is to determine which function definition applies depending on the value of x that is being input. Once that has been determined, we evaluate the function as usual by substituting  in the given value of x.

Example:

Given $$f(x) = \left\{ \begin{array}{lr} -x&\text{if}~x<0\\ x^2+1&\text{if}~x\ge0 \end{array} \right\}$$,find $$f(-1)$$, $$f(0)$$ and $$f(1)$$

Solution: First, we want to evaluate $$f$$ at $$x=-1$$, so $$f(x)=-x$$ is the equation to use

(since it applies whenever $$x<0$$).

$$\displaylines{ f(x) &=& -x \\ f(-1) &=& -(-1) \\ f(-1) &=& 1}$$

Next, we want to evaluate $$f$$ at $$x=0$$, so $$f(x)=x^2+1$$ is the equation to use.
(Since it applies whenever $$x\geq 0$$.)

$$\displaylines{f(x) &=& x^2+1 \\ f(0) &=& 0^2+1 \\ f(0) &=& 1}$$

Finally, we evaluate $$f$$ at $$x=1$$, so $$f(x)=x^2+1$$ is the equation to use. (Since it applies whenever $$x\geq 0$$.)

$$\displaylines{f(x) &=& x^2+1 \\ f(1) &=& 1^2+1 \\f(1) &=& 2}$$

## Absolute Value Functions

An absolute value function can be rewritten as a piecewise function. Absolute value is the distance from a number 'x' to 0 on the real number line. Therefore, when the value of a function is equal to zero or is positive, taking its absolute value doesn't change it; however, if the value is negative, taking absolute value changes the sign. Therefore, the definition of the function changes depending on whether or not x ≥ 0.

$$y = |x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$

Example: Write $$y = |x-2|$$ as a piecewise function and evaluate $$f(0)$$, $$f(2)$$, and $$f(5)$$.

Solution:

$$f(x) = \begin{cases} x-2, & \text{if } x \geq 0 ~or~x\geq 2 \\ -(x-2), & \text{if } x < 0~or~x<2 \end{cases}$$

First, to find $$f(0)$$, we use $$f(x) = -(x-2)$$, since this applies when $$x<2$$:

$$\displaylines{f(x) &=& -(x-2) \\ f(0) &=& -(0-2) \\ f(0) &=& 2}$$

Next, to find $$f(2)$$, we use $$f(x) = x-2$$, since this applies when $$x \geq 2$$:

$$\displaylines{f(x) &=& x-2 \\ f(2) &=& 2-2 \\ f(2) &=& 0}$$

Finally, to find $$f(5)$$, we use $$f(x) = x-2$$, since this again applies when $$x \geq 2$$:

$$\displaylines{f(x) &=& x-2 \\ f(5) &=& 5-2 \\ f(5) &=& 3}$$

Example 1:

Example 2:

Example 3: