To find the inverse of the function \( f(x) = \frac{4x - 3}{5} \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \).
\[
y = \frac{4x - 3}{5}
\]

### Step 2: Swap \( x \) and \( y \) to set up for finding the inverse.
\[
x = \frac{4y - 3}{5}
\]

### Step 3: Solve for \( y \).
First, multiply both sides by 5 to eliminate the denominator:
\[
5x = 4y - 3
\]
Then, add 3 to both sides:
\[
5x + 3 = 4y
\]
Finally, divide by 4:
\[
y = \frac{5x + 3}{4}
\]

### Step 4: Write the inverse function.
\[
f^{-1}(x) = \frac{5x + 3}{4}
\]

So, the inverse of \( f(x) = \frac{4x - 3}{5} \) is \( f^{-1}(x) = \frac{5x + 3}{4} \).

Would you like more details or clarifications on any steps?

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Here are 5 related questions you might explore:
1. What is the domain and range of the original function \( f(x) \)?
2. What is the domain and range of the inverse function \( f^{-1}(x) \)?
3. How do you verify that \( f^{-1}(x) \) is indeed the inverse of \( f(x) \)?
4. Can every function have an inverse? Why or why not?
5. How do transformations affect the inverse of a function?

**Tip**: When finding the inverse of a function, always check by composing the original and inverse functions to ensure they simplify to \( x \).

Learn how to find the inverse of the linear function f(x) = (4x - 3) / 5 using step-by-step algebraic manipulation. This problem covers inverse functions, linear equations, and how to solve for the inverse function. Suitable for high school students in Grades 9-12.

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