Question

find the inverse function of (f(x) = 16 + \sqrt3{x}).
(f^{-1}(x) =)

question 2
let (f(x) = \frac{1}{x + 15}).
(f^{-1}(x) =)

Explanation:

⚡ Using what you learned: inverse functions: definition and notation

Step 1: Solve Question 1

Set \( y = 16 + \sqrt[3]{x} \) and solve for \( x \):

$$ y = 16 + \sqrt[3]{x} $$

$$ y - 16 = \sqrt[3]{x} $$

$$ (y - 16)^3 = x $$

Interchanging \( x \) and \( y \):

$$ f^{-1}(x) = (x - 16)^3 $$

Step 2: Solve Question 2

Set \( y = \frac{1}{x + 15} \) and solve for \( x \):

$$ y = \frac{1}{x + 15} $$

$$ x + 15 = \frac{1}{y} $$

$$ x = \frac{1}{y} - 15 $$

Interchanging \( x \) and \( y \):

$$ f^{-1}(x) = \frac{1}{x} - 15 $$

Answer:

Question 1:

$$ f^{-1}(x) = (x - 16)^3 $$

Question 2:

$$ f^{-1}(x) = \frac{1}{x} - 15 $$