Question

Help answer all please wrong answers will get reported

Answer 1

If $f^{-1}(x)$ is the inverse of $f(x)$, then by definition of inverse functions,

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

Given $f(x) = 2x-8$, the inverse, if its exists, satisfies the equation above. Evaluate $f$ at the inverse, we have

$f\left(f^{-1}(x)\right) = 2 f^{-1}(x) - 8 = x$

Solve for the inverse:

$2 f^{-1}(x) - 8 = x$

$2 f^{-1}(x) = x + 8$

$\boxed{f^{-1}(x) = \dfrac12 x + 4}$

Given $g(x) = \frac13 x + 5$, we do the same thing as before to find its inverse.

$g\left(g^{-1}(x)\right) = \dfrac13 g^{-1}(x) + 5 = x \implies \boxed{g^{-1}(x) = 3x - 15}$