Find the domain and range of the inverse of the given function.
-$$f ( x ) = 5 - x ^ { 2 } ; x \geq 0$$
A) Domain: $( - \infty , 5 ]$; range: range: $[ 0 , \infty )$
B) Domain: all real numbers; range: $[ - 5 , \infty )$
C) Domain and range: all real numbers
D) Domain: all real numbers; range: $( - \infty , 5 ]$

Question

Quizplus · Accepted Answer

The original function $f(x) = 5 - x^2$ is defined for $x \geq 0$, making it a decreasing function in its domain. The range of $f(x)$ is $(-\infty, 5]$ because as $x$ increases, $5 - x^2$ decreases, but it can never be greater than 5 (when $x = 0$). For the inverse function, the domain and range swap, so the domain of the inverse is $(-\infty, 5]$ and the range is $[0, \infty)$.

Find the Domain and Range of the Inverse of the Given