Question

which function is the inverse of :
g(x) = \sqrt3{2x - 6} + 4 ?
\bigcirc f(x) = \frac{(x - 4)}{2} + 3
\bigcirc f(x) = \frac{(x - 3)^2}{2} + 3
\bigcirc f(x) = \frac{(x - 4)^3}{2} + 3
\bigcirc f(x) = \frac{(x - 4)^3}{2} + 3

Explanation:

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

We start by letting \( y = g(x) \), so we have \( y=\sqrt[3]{2x - 6}+4 \).

Step2: Solve for \( x \) in terms of \( y \)

First, subtract 4 from both sides: \( y - 4=\sqrt[3]{2x - 6} \).
Then, cube both sides to eliminate the cube root: \( (y - 4)^3=2x - 6 \).
Next, add 6 to both sides: \( (y - 4)^3+6 = 2x \).
Then, divide both sides by 2: \( x=\frac{(y - 4)^3+6}{2}=\frac{(y - 4)^3}{2}+3 \). Wait, maybe I made a miscalculation. Wait, let's re - do the steps.

Wait, original function: \( y=\sqrt[3]{2x - 6}+4 \)

Subtract 4: \( y - 4=\sqrt[3]{2x - 6} \)

Cube both sides: \( (y - 4)^3=2x - 6 \)

Add 6: \( (y - 4)^3+6 = 2x \)

Divide by 2: \( x=\frac{(y - 4)^3 + 6}{2}=\frac{(y - 4)^3}{2}+3\). Wait, but looking at the options, maybe there is a typo in my approach. Wait, maybe the original function is \( g(x)=\sqrt[3]{2(x - 3)}+4 \)? No, the original is \( g(x)=\sqrt[3]{2x - 6}+4=\sqrt[3]{2(x - 3)}+4 \).

Wait, let's try again. Let \( y=\sqrt[3]{2x - 6}+4 \)

Subtract 4: \( y - 4=\sqrt[3]{2x - 6} \)

Cube both sides: \( (y - 4)^3=2x - 6 \)

Then, \( 2x=(y - 4)^3 + 6 \)

\( x=\frac{(y - 4)^3+6}{2}=\frac{(y - 4)^3}{2}+3 \). Wait, but the options have \( \frac{(x - 4)^3}{2}+3 \) or similar. Wait, maybe the options are written with \( x \) instead of \( y \) when we swap.

After finding \( x \) in terms of \( y \), we swap \( x \) and \( y \) to get the inverse function. So the inverse function \( f(x) \) is obtained by replacing \( y \) with \( x \) in the expression for \( x \) in terms of \( y \).

So from \( x=\frac{(y - 4)^3+6}{2} \), swapping \( x \) and \( y \) gives \( y=\frac{(x - 4)^3+6}{2}=\frac{(x - 4)^3}{2}+3 \). Wait, but let's check the options. The fourth option (assuming the options are labeled as A, B, C, D and the fourth is \( f(x)=\frac{(x - 4)^3}{2}+3 \))

Wait, let's re - calculate:

\( y=\sqrt[3]{2x - 6}+4 \)

\( y - 4=\sqrt[3]{2x - 6} \)

\( (y - 4)^3=2x - 6 \)

\( 2x=(y - 4)^3+6 \)

\( x=\frac{(y - 4)^3 + 6}{2}=\frac{(y - 4)^3}{2}+3 \)

Now, replace \( y \) with \( x \): \( f(x)=\frac{(x - 4)^3}{2}+3 \)

Answer:

The inverse function is \( f(x)=\frac{(x - 4)^3}{2}+3 \) (assuming the last option is \( f(x)=\frac{(x - 4)^3}{2}+3 \))