If f(x) = 3𝑥−3x−33​ , then f -1(x) =A.3𝑥+3x+33​ B.𝑥−333x−3​ C.3(𝑥+1𝑥)3( xx+1​ )D.No inverse exists.E.𝑥+3𝑥xx+3​

Break Down the Problem

1. Identify the function: $f(x) = \frac{3x - 3}{x - 3}$.
2. Find the expression for the inverse function $f^{-1}(x)$.
3. Determine if any of the given options match the inverse function.

Relevant Concepts

The process for finding the inverse function involves the following steps:

• Replacing $f(x)$ with $y$.
• Swapping $x$ and $y$.
• Solving for $y$ to express it in terms of $x$.

Analysis and Detail

1. Start with the function: $y = \frac{3x - 3}{x - 3}$
2. Swap $x$ and $y$: $x = \frac{3y - 3}{y - 3}$
3. Multiply both sides by $(y - 3)$: $x(y - 3) = 3y - 3$ $xy - 3x = 3y - 3$
4. Rearrange the equation: $xy - 3y = 3x - 3$ $y(x - 3) = 3x - 3$
5. Solve for $y$: $y = \frac{3x - 3}{x - 3}$ Thus, the inverse function is: $f^{-1}(x) = \frac{3x - 3}{x - 3}$

Verify and Summarize

To verify, we can find $f(f^{-1}(x))$ and check if it equals $x$. After performing the calculations, we indeed find that $f(f^{-1}(x)) = x$ holds true.

None of the provided options match this derived expression. Therefore, we conclude:

Final Answer

D. No inverse exists.