Decide whether or not the functions are inverses of each other.
-$f ( x ) = \frac { 3 } { x + 2 } , g ( x ) = \frac { 2 x + 3 } { x }$
A) No
B) Yes

Question

Quizplus · Accepted Answer

To determine if two functions are inverses of each other, composing one function into the other should yield the identity function $x$. For $f(g(x))$, $$f(g(x)) = f\left(\frac{2x + 3}{x}\right) = \frac{3}{\frac{2x + 3}{x} + 2} = \frac{3x}{2x + 3 + 2x} = \frac{3x}{4x + 3}$$ which is not equal to $x$. Similarly, for $g(f(x))$, $$g(f(x)) = g\left(\frac{3}{x + 2}\right) = \frac{2\left(\frac{3}{x + 2}\right) + 3}{\frac{3}{x + 2}} = \frac{6 + 3(x + 2)}{3} = \frac{6 + 3x + 6}{3} = 2 + x$$ which is also not equal to $x$. Therefore, $f(x)$ and $g(x)$ are not inverses of each other.

Decide Whether or Not the Functions Are Inverses of Each f(x)=3x+2,g(x)=2x+3xf ( x ) = \frac { 3 } { x + 2 } , g ( x ) = \frac { 2 x + 3 } { x }f(x)=x+23​,g(x)=x2x+3​

Decide Whether or Not the Functions Are Inverses of Each $f ( x ) = \frac { 3 } { x + 2 } , g ( x ) = \frac { 2 x + 3 } { x }$