Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).
$f\(\left\)(x\(\right\))=\(\frac{x}{x-2}\)$, for $x\(\gt{2}\)$

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=x^2+1$ , then $f^{-1}\left(x\right)=\frac{1}{x^2+1}$.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

If $f\left(x\right)=\frac{1}{x}$ , then $f^{-1}\left(x\right)=\frac{1}{x}$

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).

$f\(\left\)(x\(\right\))=\(\ln\]\left\)(3x+1\(\right\))$

Find the inverse $f^{-1}\(\left\)(x\(\right\))$ of each function (on the given interval, if specified).

$f\(\left\)(x\(\right\))=10^{-2x}$

Finding inverses Find the inverse function.

ƒ(x) = 3x - 4

Finding inverses Find the inverse function.

ƒ(x) = 3x² + 1, for x ≤ 0

Inverse of composite functions

a. Let g(x) = 2x + 3 and h(x) = x³. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹

Inverse of composite functions

b. Let g(x) = x² + 1 and h(x) = √x. Consider the composite function ƒ(x) = g(h(x)). Find ƒ⁻¹ directly and then express it in terms of g⁻¹ and h⁻¹