Question

1. find the inverse of the functions below

$f(x)=\frac{1}{4}\sqrt{x - 2}-12$

Explanation:

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

\( y = \frac{1}{4}\sqrt{x - 2} - 12 \)

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

\( x = \frac{1}{4}\sqrt{y - 2} - 12 \)

Step3: Solve for \( y \), first add 12 to both sides

\( x + 12 = \frac{1}{4}\sqrt{y - 2} \)

Step4: Multiply both sides by 4

\( 4(x + 12) = \sqrt{y - 2} \)

Step5: Square both sides

\( [4(x + 12)]^2 = y - 2 \)

Step6: Simplify the left side and add 2 to both sides

\( 16(x + 12)^2 + 2 = y \)

Step7: Replace \( y \) with \( f^{-1}(x) \)

\( f^{-1}(x) = 16(x + 12)^2 + 2 \) (with the domain consideration that \( x \geq -12 \) since the original function's range is \( y \geq -12 \))

Answer:

The inverse function is \( f^{-1}(x) = 16(x + 12)^2 + 2 \) (for \( x \geq -12 \))