Find the inverse function of f.
f(x) = 
7x
x − 2


To find the inverse of the function \( f(x) = \frac{7x}{x - 2} \), follow these steps:

### Step 1: Replace \( f(x) \) with \( y \).
\[
y = \frac{7x}{x - 2}
\]

### Step 2: Swap \( x \) and \( y \) to reflect the inverse function.
\[
x = \frac{7y}{y - 2}
\]

### Step 3: Solve for \( y \).
First, eliminate the denominator by multiplying both sides by \( y - 2 \):
\[
x(y - 2) = 7y
\]
Expand the left-hand side:
\[
xy - 2x = 7y
\]
Move all terms involving \( y \) to one side:
\[
xy - 7y = 2x
\]
Factor out \( y \) on the left side:
\[
y(x - 7) = 2x
\]
Finally, solve for \( y \):
\[
y = \frac{2x}{x - 7}
\]

### Step 4: Write the inverse function.
The inverse of \( f(x) \) is:
\[
f^{-1}(x) = \frac{2x}{x - 7}
\]

Would you like more details or have any questions on this process?

Here are 5 related questions:
1. How can you verify if two functions are inverses of each other?
2. What is the domain and range of \( f(x) \) and its inverse?
3. How does the horizontal line test relate to finding an inverse function?
4. How would you graph both \( f(x) \) and \( f^{-1}(x) \) on the same coordinate plane?
5. What happens if a function is not one-to-one? Can it still have an inverse?

**Tip:** Always check that the function is one-to-one before finding its inverse to ensure it has a valid inverse function.

Find the inverse function of f. f(x) = 7x / (x − 2)

Discover how to find the inverse of the rational function f(x) = 7x / (x − 2) with a step-by-step solution. This algebraic problem highlights key concepts like inverse functions, rational functions, and the inverse function theorem. Suitable for students in Grades 10-12.

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