Question

inverses quiz review
name:
period:
part 4: determine the inverse relationship for each function. explain how you
determined your answer.
a) ( f(x) = 3x - 8 )
the function ( f(x) ) is described by starting with
some number ( x ), multiplying it by 3, and then
subtracting 8 from that quantity. i found the inverse
( f^{-1}(x) ) by reversing the order of the steps and
performing the inverse operations (multiply by 3→divide by 3,
subtract 8→add 8). ( \boldsymbol{f^{-1}(x) = \frac{x + 8}{3}} ) (circled)
b) ( g(x) = \frac{x + 5}{4} )
c) ( h(x) = sqrt3{x - 9} )
(handwritten notes: \steps\, \what i did with that info\)

Explanation:

Part 4: Determine the inverse relationship for each function. Explain how you determined your answer.

a) \( f(x) = 3x - 8 \)

Step 1: Replace \( f(x) \) with \( y \)

\( y = 3x - 8 \)

Step 2: Swap \( x \) and \( y \)

\( x = 3y - 8 \)

Step 3: Solve for \( y \)

Add 8 to both sides: \( x + 8 = 3y \)
Divide both sides by 3: \( y = \frac{x + 8}{3} \)

Step 4: Replace \( y \) with \( f^{-1}(x) \)

\( f^{-1}(x) = \frac{x + 8}{3} \)

Step 1: Replace \( g(x) \) with \( y \)

\( y = \frac{x + 5}{4} \)

Step 2: Swap \( x \) and \( y \)

\( x = \frac{y + 5}{4} \)

Step 3: Solve for \( y \)

Multiply both sides by 4: \( 4x = y + 5 \)
Subtract 5 from both sides: \( y = 4x - 5 \)

Step 4: Replace \( y \) with \( g^{-1}(x) \)

\( g^{-1}(x) = 4x - 5 \)

Step 1: Replace \( h(x) \) with \( y \)

\( y = \sqrt[3]{x - 9} \)

Step 2: Swap \( x \) and \( y \)

\( x = \sqrt[3]{y - 9} \)

Step 3: Solve for \( y \)

Cube both sides: \( x^3 = y - 9 \)
Add 9 to both sides: \( y = x^3 + 9 \)

Step 4: Replace \( y \) with \( h^{-1}(x) \)

\( h^{-1}(x) = x^3 + 9 \)

Answer:

\( f^{-1}(x) = \frac{x + 8}{3} \) (Explanation: To find the inverse, we swap \( x \) and \( y \) in the original function \( y = 3x - 8 \), then solve for \( y \) by reversing the operations: add 8 to \( x \) and then divide by 3.)

b) \( g(x) = \frac{x + 5}{4} \)