Let  f (x) = 2 + x + ln(x − 9),  x > 9.

(a)	Find f −1(12)
(b)	Find f −1( f (22))


We are given the function \( f(x) = 2 + x + \ln(x - 9) \) with \( x > 9 \).

### Part (a): Find \( f^{-1}(12) \)

To find \( f^{-1}(12) \), we need to determine the value of \( x \) such that:

\[
f(x) = 12
\]

That is,

\[
2 + x + \ln(x - 9) = 12
\]

#### Step 1: Isolate the logarithmic expression

Subtract 2 from both sides:

\[
x + \ln(x - 9) = 10
\]

#### Step 2: Solve for \( x \)

We can't solve this algebraically, so we will solve it numerically.

---

### Part (b): Find \( f^{-1}(f(22)) \)

By the definition of inverse functions, \( f^{-1}(f(x)) = x \). Thus,

\[
f^{-1}(f(22)) = 22
\]

So, \( f^{-1}(f(22)) = 22 \) without further calculation.

---

Let's now solve part (a) numerically. I'll compute the value of \( x \) for which \( x + \ln(x - 9) = 10 \).For part (a), the solution to \( f(x) = 12 \) is \( x = 10 \). Therefore:

\[
f^{-1}(12) = 10
\]

To summarize:
- \( f^{-1}(12) = 10 \)
- \( f^{-1}(f(22)) = 22 \)

Do you need further details or have any questions?

Here are 5 related questions for deeper understanding:
1. What is the general process for finding the inverse of a function?
2. How can logarithmic equations like this one be solved algebraically or numerically?
3. Why is the domain restriction \( x > 9 \) necessary for this function?
4. How would you verify the inverse of a function graphically?
5. What are the practical applications of finding inverse functions in real-world problems?

**Tip:** In problems involving inverse functions, applying \( f(f^{-1}(x)) = x \) and \( f^{-1}(f(x)) = x \) can often simplify your work.

Let f(x) = 2 + x + ln(x − 9), x > 9.
(a) Find f−1(12)
(b) Find f−1(f(22))

Explore the solution to the inverse function problem for f(x) = 2 + x + ln(x - 9). Learn how to find f−1(12) using numerical methods and apply the inverse function theorem to solve f−1(f(22)). Suitable for students in Grades 10-12.

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