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Let's break down the steps needed to solve this problem set:

### Given:
- \( f(x) = \frac{3(x+1)}{2x^2 + 7x - 4} - \frac{1}{x + 4}, \ x \in \mathbb{R}, \ x > \frac{1}{2} \)
- \( g(x) = \ln(x+1), \ x > -1 \)

### (a) Show that \( f(x) = \frac{1}{2x - 1} \)
This step requires simplifying \( f(x) \).

1. **Step 1:** Simplify \( \frac{3(x+1)}{2x^2 + 7x - 4} \).
   - Factorize the quadratic denominator if possible.
   
2. **Step 2:** Combine the two terms of \( f(x) \) into one expression.

### (b) Find \( f^{-1}(x) \)
To find the inverse function of \( f(x) \), follow these steps:
1. Set \( y = f(x) = \frac{1}{2x - 1} \).
2. Swap \( y \) and \( x \) to find the inverse.
3. Solve for \( y \) in terms of \( x \).

### (c) Find the domain of \( f^{-1}(x) \)
Since the inverse function is \( f^{-1}(x) \), find the domain by considering the range of the original function \( f(x) \).

### (d) Find \( g^{-1}(x) \) and its domain
Since \( g(x) = \ln(x + 1) \), follow these steps to find its inverse:
1. Set \( y = g(x) = \ln(x + 1) \).
2. Solve for \( x \) in terms of \( y \).

The domain of \( g^{-1}(x) \) corresponds to the range of \( g(x) \).

### (e) Sketch the curves of \( y = g(x) \) and \( y = g^{-1}(x) \)
1. Plot the logarithmic curve \( y = \ln(x + 1) \).
2. Plot the inverse curve.
3. Identify the points where the curves cross the axes.

### (f) Solve \( f(g(x)) = \frac{1}{7} \)
1. Use the composite function \( f(g(x)) \).
2. Set \( f(g(x)) = \frac{1}{7} \) and solve for \( x \) in terms of \( e \).

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Do you want me to show the detailed solution for each part one by one?

The function f and g are defined by the equations f(x) = 3(x+1)/(2x^2+7x-4) - 1/(x+4) for x > 1/2 and g(x) = ln(x + 1) for x > -1. Solve the following parts: (a) Show that f(x) = 1/(2x - 1). (b) Find f^(-1)(x). (c) Find the domain of f^(-1)(x). (d) Find g^(-1)(x) and its domain. (e) Sketch the curves for y = g(x) and y = g^(-1)(x). (f) Solve f(g(x)) = 1/7.

Solve problems involving functions, their inverses, and composite functions. Given f(x) = 3(x+1)/(2x^2+7x-4) - 1/(x+4) and g(x) = ln(x + 1), find their inverses, domains, and sketch curves for both y = g(x) and y = g^(-1)(x). Suitable for Grades 11-12.

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