Question

this question has two parts. first, answer part a. then, answer part b.
part a
structure write the inverse of each function.
a. $f(x) = \frac{x - 10}{3}$

part b
b. $g(x) = \frac{3}{4}x + 6$

Explanation:

Part A: Step1: Replace \( f(x) \) with \( y \)

\( y = \frac{x - 10}{3} \)

Part A: Step2: Swap \( x \) and \( y \)

\( x = \frac{y - 10}{3} \)

Part A: Step3: Solve for \( y \)

Multiply both sides by 3: \( 3x = y - 10 \)
Add 10 to both sides: \( y = 3x + 10 \)
So, \( f^{-1}(x) = 3x + 10 \)

Part B: Step1: Replace \( g(x) \) with \( y \)

\( y = \frac{3}{4}x + 6 \)

Part B: Step2: Swap \( x \) and \( y \)

\( x = \frac{3}{4}y + 6 \)

Part B: Step3: Solve for \( y \)

Subtract 6 from both sides: \( x - 6 = \frac{3}{4}y \)
Multiply both sides by \( \frac{4}{3} \): \( y = \frac{4}{3}(x - 6) = \frac{4}{3}x - 8 \)
So, \( g^{-1}(x) = \frac{4}{3}x - 8 \)

Answer:

Part A: \( \boldsymbol{f^{-1}(x) = 3x + 10} \)
Part B: \( \boldsymbol{g^{-1}(x) = \frac{4}{3}x - 8} \)