91Ó°ÊÓ

Function evaluation is a process that allows you to find the output of a function given an input value. For piecewise functions, you first identify which piece of the function applies to your input. Next, you substitute the input value into the identified expression.
We can illustrate this with an exercise: If we have a piecewise function like this:

\[f(x)=\left\{\begin{array}{ll}-5 x-8, & \text { for } x<-2 \frac{1}{2} x+5, & \text { for }-2 \leq x \leq 4 \10-2 x, & \text { for } x>4\end{array}\right.\]

Let's compute \( f(-4), f(-2), f(4), \text{ and } f(6) \).

For \( f(-4) \): Since \( -4 < -2 \), we use \( -5x-8 \). Substituting \( -4 \) into the function: \( f(-4) = -5(-4) - 8 = 20 - 8 = 12 \).
For \( f(-2) \): Since \( -2 \leq x \leq 4 \), we use \( \frac{1}{2}x + 5 \). Substituting \( -2 \) into the function: \( f(-2) = \frac{1}{2}(-2) + 5 = -1 + 5 = 4 \).
For \( f(4) \): Since \( -2 \leq x \leq 4 \), we use \( \frac{1}{2}x + 5 \). Substituting \( 4 \) into the function: \( f(4) = \frac{1}{2}(4) + 5 = 2 + 5 = 7 \).
For \( f(6) \): Since \( x > 4 \), we use \( 10-2x \). Substituting \( 6 \) into the function: \( f(6) = 10 - 2(6) = 10 - 12 = -2 \).

The domain of a function is the complete set of possible input values (x-values) for which the function is defined. For piecewise functions, these values can be broken into segments.
The range of the function is the set of possible output values (y-values) that the function can produce.
Let's apply this to our piecewise function example.
\[f(x)=\left\{\begin{array}{ll}-5 x-8, & \text { for } x<-2 \frac{1}{2} x+5, & \text { for }-2 \leq x \leq 4 \10-2 x, & \text { for } x>4\end{array}\right.\]
The domain here covers all real numbers because there are no restrictions on \( x \). The specific segments are:
- \( x < -2 \)
- \( -2 \leq x \leq 4 \)
- \( x > 4 \)
The ranges depend on each segment.
- For \( x < -2 \), solving \( -5x-8 \) gives us values from \( -fty \text{ to } fty \).
- For \( -2 \leq x \leq 4 \), solving \( \frac{1}{2}x + 5 \) gives values from \( -1 \text{ to } 7 \).
- For \( x > 4 \), values of \( 10 - 2x \) range from values greater \( -fty \text{ to } ewline \) as \( x \rightarrow fty \) .

The substitution method is a way to evaluate functions by replacing the variable (usually \( x \)) with a specific value. This method is especially useful for piecewise functions where you need to evaluate different expressions based on the input values.
For our example:
\[f(x)=\left\{\begin{array}{ll}-5 x-8, & \text { for } x<-2 \frac{1}{2} x+5, & \text { for }-2 \leq x \leq 4 \10-2 x, & \text { for } x>4\end{array}\right.\]
This means:

• If \( x = -4 \), substitute -4 into \( -5x-8 \) to get 12.
• If \( x = -2 \), substitute -2 into \( \frac{1}{2}x + 5 \) to get 4.
• If \( x = 4 \), substitute 4 into \( \frac{1}{2}x + 5 \) to get 7.
• If \( x = 6 \), substitute 6 into \( 10 - 2x \) to get -2.

This method allows you to effectively evaluate each piece of a complicated function, simplifying the process into manageable steps.