Decide whether or not the functions are inverses of each other.
-$$f ( x ) = \frac { 1 } { x + 6 } , g ( x ) = \frac { 6 x + 1 } { x }$$
A) Yes
B) Yes; Exclude the value $\{ - 6 \}$
C) No

Question

Quizplus · Accepted Answer

To determine if two functions are inverses of each other, we can compose them (i.e., substitute one function into the other) and see if the result is the identity function, $x$. First, compose $f(g(x))$:$$f(g(x)) = f\left(\frac{6x + 1}{x}\right) = \frac{1}{\frac{6x + 1}{x} + 6} = \frac{1}{\frac{6x + 1 + 6x}{x}} = \frac{1}{\frac{12x + 1}{x}} = \frac{x}{12x + 1}$$Next, compose $g(f(x))$:$$g(f(x)) = g\left(\frac{1}{x + 6}\right) = \frac{6\left(\frac{1}{x + 6}\right) + 1}{\frac{1}{x + 6}} = \frac{6 + x + 6}{1} = x + 12$$Neither composition results in the identity function $x$, indicating that $f(x)$ and $g(x)$ are not inverses of each other.

Decide Whether or Not the Functions Are Inverses of Each f(x)=1x+6,g(x)=6x+1xf ( x ) = \frac { 1 } { x + 6 } , g ( x ) = \frac { 6 x + 1 } { x }f(x)=x+61​,g(x)=x6x+1​

Decide Whether or Not the Functions Are Inverses of Each $f ( x ) = \frac { 1 } { x + 6 } , g ( x ) = \frac { 6 x + 1 } { x }$