Question

the domain of the piecewise function is (- ∞, ∞)
a. graph the function.
f(x)=
\

$$\begin{cases} 0 & \\text{if } x \\leq 5 \\\\ -8 & \\text{if } x > 5 \\end{cases}$$

b. use your graph to determine the function’s range
b. what is the range of the entire piecewise function? select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
\bigcirc a the range does not have any isolated values. it can be described by \square (type your answer in interval notation)
\bigcirc b the range has at least one isolated value. it can be described as the union of the interval(s) \square and the set \square (use a comma to separate answers as needed)
\bigcirc c the range consists exclusively of one or more isolated values. it can be described as \square (use a comma to separate answers as needed)

Explanation:

Step1: Analyze the Piecewise Function

The function is \( f(x) =

$$\begin{cases} 0 & \text{if } x \leq 5 \\ -8 & \text{if } x > 5 \end{cases}$$

\). So for all \( x \leq 5 \), \( f(x) = 0 \), and for all \( x > 5 \), \( f(x) = -8 \).

Step2: Determine the Range

The range is the set of all possible output values (\( y \)-values). From the function, the possible \( y \)-values are \( 0 \) (when \( x \leq 5 \)) and \( -8 \) (when \( x > 5 \)). So the range consists of two isolated values: \( -8 \) and \( 0 \).

Answer:

The range is \(\{-8, 0\}\) (using set notation as the range has at least one isolated value, so option B applies and the range is the set of these two values).