Question

which function, ( g^{-1}(x) ), represents the inverse of the function ( g(x) = -\frac{1}{3}(x - 6) + 5 )?

( \bigcirc ) ( g^{-1}(x) = -3x - 7 )
( \bigcirc ) ( g^{-1}(x) = -3x + 21 )
( \bigcirc ) ( g^{-1}(x) = -\frac{1}{3}x + 1 )
( \bigcirc ) ( g^{-1}(x) = -\frac{1}{3}x + 7 )

Explanation:

Step1: Start with the function

Given \( g(x) = -\frac{1}{3}(x - 6)+5 \), first simplify it.
\( g(x)=-\frac{1}{3}x + 2 + 5=-\frac{1}{3}x + 7 \)

Step2: Swap x and y

Let \( y = g(x) \), so \( x = -\frac{1}{3}y + 7 \)

Step3: Solve for y

Subtract 7: \( x - 7 = -\frac{1}{3}y \)
Multiply by -3: \( -3(x - 7)=y \)
Simplify: \( y=-3x + 21 \)
So \( g^{-1}(x)=-3x + 21 \)

Answer:

\( g^{-1}(x)=-3x + 21 \) (the third option, e.g., if options are labeled as first, second, third, fourth, then the third option: \( g^{-1}(x)=-3x + 21 \))