f(x)= 
5
x
​
 +5, find f, to the power minus 1 , left parenthesis, x, right parenthesisf 
−1
 (x)

To find the inverse function, \( f^{-1}(x) \), of the given function:

\[
f(x) = \frac{5}{x} + 5
\]

### Step-by-step process:

1. **Start by replacing \( f(x) \) with \( y \):**

\[
y = \frac{5}{x} + 5
\]

2. **Swap \( x \) and \( y \) to find the inverse:**

\[
x = \frac{5}{y} + 5
\]

3. **Solve for \( y \):**
   - Subtract 5 from both sides:

\[
x - 5 = \frac{5}{y}
\]

   - Multiply both sides by \( y \):

\[
y(x - 5) = 5
\]

   - Solve for \( y \) by dividing both sides by \( x - 5 \):

\[
y = \frac{5}{x - 5}
\]

Thus, the inverse function is:

\[
f^{-1}(x) = \frac{5}{x - 5}
\]

Would you like further details or have any questions?

Here are 5 related questions for practice:
1. If \( f(x) = \frac{5}{x+1} + 2 \), what is \( f^{-1}(x) \)?
2. What is the domain of \( f^{-1}(x) = \frac{5}{x - 5} \)?
3. How can you verify that \( f(f^{-1}(x)) = x \) for the function found?
4. If \( f(x) = \frac{a}{x} + b \), how do you generally find \( f^{-1}(x) \)?
5. What are the intercepts of the inverse function \( f^{-1}(x) \)?

**Tip:** To find the inverse of a function, swap \( x \) and \( y \) and then solve for \( y \).

Learn how to find the inverse of the function f(x) = 5/x + 5 with a clear, step-by-step solution. This problem explores inverse functions, algebraic manipulation, and function swapping. Suitable for students in Grades 9-11.

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