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, we check if $f(g(x)) = x$ and $g(f(x)) = x$. Here, $f(g(x)) = 5(\sqrt{\frac{x - 1}{5}})^2 + 1 = 5(\frac{x - 1}{5}) + 1 = x - 1 + 1 = x$, and $g(f(x)) = \sqrt{\frac{5x^2 + 1 - 1}{5}} = \sqrt{x^2} = |x|$. Since $g(f(x))$ returns $|x|$, it only equals $x$ for $x \geq 0$, implying that $f(x)$ and $g(x)$ are inverses only if we exclude the interval where $x < 0$. However, the correct exclusion to ensure $f(x)$ is a function (due to the square root in $g(x)$) is for inputs to $g(x)$ that would result from $f(x)$ outputs less than 1, hence the exclusion of $(-\infty, 1)$ for the domain of $g(x)$.

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 $f ( x ) = 5 x ^ { 2 } + 1 , g ( x ) = \sqrt { \frac { x - 1 } { 5 } }$