Let f be a real valued invertible function such that f((2x - 3)/(x - 2)) = 5x - 2 x ≠ 2. Then value of f(13) is___

Given that \( f\left(\frac{2x - 3}{x - 2}\right) = 5x - 2 \), we need to find the value of \( f(13) \).

### Step 1: Express the argument of \( f \) in terms of \( y \).

Let \( y = \frac{2x - 3}{x - 2} \). We need to solve for \( x \) in terms of \( y \):

\[
y(x - 2) = 2x - 3
\]

\[
yx - 2y = 2x - 3
\]

\[
yx - 2x = 2y - 3
\]

\[
x(y - 2) = 2y - 3
\]

\[
x = \frac{2y - 3}{y - 2}
\]

### Step 2: Substitute \( x \) into the given equation.

Now, substitute \( x = \frac{2y - 3}{y - 2} \) into the original function:

\[
f\left(y\right) = 5\left(\frac{2y - 3}{y - 2}\right) - 2
\]

Simplify the expression inside the function:

\[
f(y) = \frac{10y - 15}{y - 2} - 2
\]

Simplify further:

\[
f(y) = \frac{10y - 15 - 2(y - 2)}{y - 2}
\]

\[
f(y) = \frac{10y - 15 - 2y + 4}{y - 2}
\]

\[
f(y) = \frac{8y - 11}{y - 2}
\]

### Step 3: Evaluate \( f(13) \).

Now, set \( y = 13 \):

\[
f(13) = \frac{8(13) - 11}{13 - 2}
\]

\[
f(13) = \frac{104 - 11}{11}
\]

\[
f(13) = \frac{93}{11}
\]

Thus, the value of \( f(13) \) is \(\boxed{\frac{93}{11}}\). 

Would you like to see further details, or have any questions?

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Here are some related questions for you:
1. What would be the inverse function \( f^{-1}(x) \)?
2. How do you determine if a function is invertible?
3. What role does the domain of \( f(x) \) play in finding the inverse?
4. What happens to the range of \( f(x) \) when the function is inverted?
5. Can you find the derivative of \( f(x) \) for general \( x \)?
6. How do you check if a function is one-to-one?
7. What is the significance of the condition \( x \neq 2 \) in the original function?
8. How would the problem change if \( f(x) \) were not invertible?

**Tip**: When solving for the inverse of a function, always double-check the algebraic steps to ensure the expression is simplified correctly.

Explore how to find f(13) for an invertible function using a transformation equation approach. Understand the steps involving linear equations and domain considerations. Suitable for advanced high school students and those studying function transformations.

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