Let us solve the question step by step.

Part (a): Show that $f(x)$ and $g(x)$ are inverse functions

To verify that $f(x)$ and $g(x)$ are inverses, we need to show that:

1. $f(g(x)) = x$
2. $g(f(x)) = x$

Given functions:

• $f(x) = e^{3 - 2x}$
• $g(x) = \frac{3 - \ln x}{2}$

Step 1: Calculate $f(g(x))$: $f(g(x)) = f\left(\frac{3 - \ln x}{2}\right) = e^{3 - 2 \cdot \frac{3 - \ln x}{2}}$ Simplify: $e^{3 - (3 - \ln x)} = e^{\ln x} = x$

Step 2: Calculate $g(f(x))$: $g(f(x)) = g\left(e^{3 - 2x}\right) = \frac{3 - \ln\left(e^{3 - 2x}\right)}{2}$ Simplify: $\ln\left(e^{3 - 2x}\right) = 3 - 2x \quad \text{so:} \quad g(f(x)) = \frac{3 - (3 - 2x)}{2} = \frac{2x}{2} = x$

Thus, $f$ and $g$ are inverses of each other.

Part (b): Simplify $(f \circ g)(x)$

We already calculated $f(g(x))$ in part (a): $f(g(x)) = x$ Thus, $(f \circ g)(x) = x$.

Part (c): Sketch the graphs of $f(x)$ and $g(x)$

To sketch the graphs of $f(x) = e^{3 - 2x}$ and $g(x) = \frac{3 - \ln x}{2}$, we need:

1. Intercepts:

   • For $f(x)$: $\text{When } x = 0, f(0) = e^{3 - 2(0)} = e^3 \quad \text{so the y-intercept is } (0, e^3).$
   • For $g(x)$: $\text{When } x = 1, g(1) = \frac{3 - \ln(1)}{2} = \frac{3 - 0}{2} = \frac{3}{2}.$ So, $g(x)$ has a y-intercept of $(1, \frac{3}{2})$.

2. Asymptotes:

   • For $f(x) = e^{3 - 2x}$: $\text{As } x \to \infty, f(x) \to 0 \quad \text{(horizontal asymptote at } y = 0\text{)}.$
   • For $g(x) = \frac{3 - \ln x}{2}$: $\text{As } x \to 0^+, \ln x \to -\infty, \quad g(x) \to \infty.$ $\text{As } x \to \infty, \ln x \to \infty, \quad g(x) \to -\infty.$

3. Behavior:

   • $f(x)$ is an exponential decay function.
   • $g(x)$ is a logarithmic function reflected and shifted.

Part (d): Number of solutions to $f(x) = g(x)$

To solve $f(x) = g(x)$, we equate the two functions: $e^{3 - 2x} = \frac{3 - \ln x}{2}$ This equation cannot be solved algebraically, but we can analyze the graphs to determine the number of intersections.

• $f(x) = e^{3 - 2x}$ is strictly decreasing.
• $g(x) = \frac{3 - \ln x}{2}$ decreases for $x > 1$.

There will likely be one solution as the graphs intersect only once based on their behavior.

Would you like a detailed graph or further clarification on any part?

Questions to Expand Understanding:

1. How can you use the derivative to confirm that $f(x)$ and $g(x)$ intersect only once?
2. What happens to the inverses of $f(x)$ and $g(x)$ when transformations are applied?
3. Can you calculate the intersection point $f(x) = g(x)$ numerically?
4. Why is $\ln x$ undefined for $x \leq 0$, and how does this impact $g(x)$?
5. How do horizontal asymptotes affect the behavior of $f(x)$?

Tip:

For exponential and logarithmic functions, always check their domains and asymptotes carefully before solving equations or graphing.