https://file.aiquickdraw.com/mathbot/file/1725555750266vpdk6m7w.jpg

The problem from the image can be broken down as follows:

### Problem 12
We are given the function:
\[ f(x) = \ln(x-1) + \ln 3, \text{ for } x > 1. \]

### Part (a)
- **Find \( f(x) \).**
- Let \( g(x) = e^x \text{ for } x \in \mathbb{R} \).

### Part (b)
- **Find \( g \circ f(x) \),** giving your answer in the form \( ax + b \), where \( a, b \in \mathbb{Z} \).
- Find the domain and range of \( g \circ f \).

---

### Step-by-Step Solution

#### Part (a)
The given function is:
\[
f(x) = \ln(x-1) + \ln 3.
\]
Using the properties of logarithms, particularly the product rule \( \ln a + \ln b = \ln(ab) \), we can rewrite this as:
\[
f(x) = \ln(3(x-1)).
\]
Thus, \( f(x) = \ln(3(x-1)) \).

#### Part (b)
The function \( g(x) = e^x \). Now, to find \( g \circ f(x) \), we calculate:
\[
g(f(x)) = g(\ln(3(x-1))) = e^{\ln(3(x-1))}.
\]
Since \( e^{\ln y} = y \), this simplifies to:
\[
g(f(x)) = 3(x-1).
\]
Expanding this, we get:
\[
g(f(x)) = 3x - 3.
\]
Thus, \( g(f(x)) = 3x - 3 \), which is in the form \( ax + b \) where \( a = 3 \) and \( b = -3 \).

#### Domain and Range of \( g \circ f \)
- The domain of \( f(x) = \ln(3(x-1)) \) requires \( x > 1 \), so the domain of \( g \circ f(x) \) is \( x > 1 \).
- The range of \( g(f(x)) = 3x - 3 \) is all real numbers since it is a linear function, which can take any real value as \( x \) increases.

### Final Answers:
- \( f(x) = \ln(3(x-1)) \)
- \( g(f(x)) = 3x - 3 \), with \( a = 3 \) and \( b = -3 \)
- The domain of \( g \circ f \) is \( x > 1 \), and the range is \( \mathbb{R} \).

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Let me know if you want further clarifications or details. Here are 5 related questions for further practice:

1. Can you simplify the expression \( \ln(a) + \ln(b) - \ln(c) \)?
2. How do you find the inverse of a function like \( f(x) = \ln(3x - 2) \)?
3. What is the derivative of \( f(x) = \ln(2x + 5) \)?
4. If \( g(x) = e^x \), what is the derivative of \( g(f(x)) \) where \( f(x) = \ln(x-1) \)?
5. How do you find the domain of \( f(x) = \ln(x^2 - 4) \)?

**Tip:** Always remember the logarithmic identities like the product rule, quotient rule, and power rule when simplifying or transforming logarithmic functions.

Let f(x) = ln(x - 1) + ln 3, for x > 1. a) Find f(x). Let g(x) = e^x for x ∈ ℝ. b) Find g(f(x)), giving your answer in the form ax + b, where a, b ∈ ℤ. Find the domain and range of g(f).

Explore the logarithmic and exponential functions in this problem, where you find f(x) = ln(x - 1) + ln 3 and g(f(x)) in the form ax + b. The solution includes a step-by-step guide to applying logarithmic rules and determining the domain and range of the composition. Ideal for Grades 11-12.

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