Question

find the inverse for each of the following functions.

(f(x) = 3x + 12)
(f^{-1}(x) =)

(g(x) = 15x^3 - 10)
(g^{-1}(x) =)

(h(x) = \frac{3}{x + 10})
(h^{-1}(x) =)

(j(x) = \sqrt3{x + 15})
(j^{-1}(x) =)

Explanation:

Step1: Find the inverse of \(f(x)\)

$$y = 3x + 12 \implies x = 3y + 12 \implies y = \frac{x - 12}{3}$$

Step2: Find the inverse of \(g(x)\)

$$y = 15x^3 - 10 \implies x = 15y^3 - 10 \implies y = \sqrt[3]{\frac{x + 10}{15}}$$

Step3: Find the inverse of \(h(x)\)

$$y = \frac{3}{x + 10} \implies x = \frac{3}{y + 10} \implies y = \frac{3}{x} - 10$$

Step4: Find the inverse of \(j(x)\)

$$y = \sqrt[3]{x + 15} \implies x = \sqrt[3]{y + 15} \implies y = x^3 - 15$$

Answer:

\(f^{-1}(x) = \frac{x - 12}{3}\)
\(g^{-1}(x) = \sqrt[3]{\frac{x + 10}{15}}\)
\(h^{-1}(x) = \frac{3}{x} - 10\)
\(j^{-1}(x) = x^3 - 15\)