Question

Determining if two functions are inverses. 15 pts

Answer 1

An inverse function $f^{-1}$ is such that

$f\left(f^{-1}(x)\right) = f^{-1}(f(x)) = x$

(a) Given $f(x) = x-3$ and $g(x) = x+3$, we have

$f(g(x)) = f(x+3) = (x+3)-3 = x$

and

$g(f(x)) = g(x-3) = (x-3)+3 = x$

so $f$ and $g$ are indeed inverses of one another.

(b) Given $f(x)=\frac1{4x}$ and $g(x)=-\frac1{4x}$, we have

$f(g(x)) = f\left(-\dfrac1{4x}\right) = \dfrac1{4\left(-\frac1{4x}\right)} = -\dfrac1{\frac1x} = -x \neq x$

so $f$ and $g$ are not inverses of one another.

Answer 2

The answer is the first one because u could see how it is but I think I try my best