Decide whether or not the functions are inverses of each other.
-$$f ( x ) = 5 x ^ { 2 } + 1 , g ( x ) = \sqrt { \frac { x - 1 } { 5 } }$$
A) Yes; Exclude the interval $( - \infty , 1 )$
B) No
C) Yes; No values need to be excluded.
D) Yes; Exclude the interval $( - \infty , 5 )$

Question

Quizplus · Accepted Answer

To determine if two functions are inverses of each other, we can compose them (i.e., substitute one into the other) and see if the result is the identity function $x$. First, compose $g(f(x))$:$$g(f(x)) = g(5x^2 + 1) = \sqrt{\frac{(5x^2 + 1) - 1}{5}} = \sqrt{x^2} = |x|$$Since $g(f(x)) = |x|$, the composition does not directly yield $x$, but $|x|$, which means the functions are inverses of each other only for $x \geq 0$. This is because the square root function, by definition, returns the non-negative square root. Therefore, the domain of $g(x)$ must be restricted to ensure it is a proper inverse of $f(x)$. Given $f(x) = 5x^2 + 1$, its range is $[1, \infty)$ because the minimum value of $5x^2$ is 0 (when $x = 0$), and adding 1 shifts the entire graph up by 1. Since $g(x)$ is supposed to be the inverse of $f(x)$, its domain should match the range of $f(x)$, which is $[1, \infty)$. Therefore, we exclude the interval $(- \infty, 1)$ to ensure that $g(x)$ properly inverts $f(x)$ for its entire domain, making choice A correct.

Decide Whether or Not the Functions Are Inverses of Each f(x)=5x2+1,g(x)=x−15f ( x ) = 5 x ^ { 2 } + 1 , g ( x ) = \sqrt { \frac { x - 1 } { 5 } }f(x)=5x2+1,g(x)=5x−1​​

Decide Whether or Not the Functions Are Inverses of Each $f ( x ) = 5 x ^ { 2 } + 1 , g ( x ) = \sqrt { \frac { x - 1 } { 5 } }$