The step function $g(x)$ is defined as shown.
$g(x)=\left\{\begin{array}{ll}
-3, & x \leq 0 \\
2, & 0
5, & 4
\end{array}\right.$
What is the range of $g(x)$ ?
A. $-3 \leq g(x) \leq 5$
B. $0 \leq g(x) \leq 10$
C. $\{-3,2,5\}$
D. $\{0,4,10\}$
Answer (1)
Identify the distinct values of the step function g ( x ) for each interval.
For x ≤ 0 , g ( x ) = − 3 .
For 0 < x ≤ 4 , g ( x ) = 2 .
For 4 < x ≤ 10 , g ( x ) = 5 .
The range of g ( x ) is { − 3 , 2 , 5 } .
Explanation
Understanding the Step Function The function g ( x ) is a step function, which means it has constant values over specific intervals. We need to identify all the distinct values that g ( x ) takes to determine its range.
Value for x <= 0 For x l e q 0 , g ( x ) = − 3 . So, − 3 is in the range of g ( x ) .
Value for 0 < x <= 4 For 0 < x ≤ 4 , g ( x ) = 2 . So, 2 is in the range of g ( x ) .
Value for 4 < x <= 10 For 4 < x ≤ 10 , g ( x ) = 5 . So, 5 is in the range of g ( x ) .
Determining the Range Therefore, the range of g ( x ) is the set of all distinct values that g ( x ) takes, which is { − 3 , 2 , 5 } .
Examples
Step functions are used to model situations where the output value changes abruptly at certain points. For example, the cost of parking in a garage might be modeled by a step function: it costs a fixed amount for the first hour, then a different fixed amount for the second hour, and so on. Understanding the range of such a function helps you know all the possible costs you might incur.
