Let’s solve the given problems step by step:

(a) Find $f(g(x))$

We need to substitute $g(x) = -\frac{1}{2}(x - 3)$ into $f(x) = -2x + 3$.

1. Substitute $g(x)$ into $f(x)$: $f(g(x)) = f\left(-\frac{1}{2}(x - 3)\right)$

2. Replace $x$ in $f(x) = -2x + 3$ with $-\frac{1}{2}(x - 3)$: $f(g(x)) = -2\left(-\frac{1}{2}(x - 3)\right) + 3$

3. Simplify: $f(g(x)) = (-2) \cdot \left(-\frac{1}{2}(x - 3)\right) + 3$ $f(g(x)) = (1)(x - 3) + 3$ $f(g(x)) = x - 3 + 3$ $f(g(x)) = x$

Thus, $f(g(x)) = x$.

(b) Find $g(f(x))$

Now, substitute $f(x) = -2x + 3$ into $g(x) = -\frac{1}{2}(x - 3)$.

1. Substitute $f(x)$ into $g(x)$: $g(f(x)) = g\left(-2x + 3\right)$

2. Replace $x$ in $g(x) = -\frac{1}{2}(x - 3)$ with $-2x + 3$: $g(f(x)) = -\frac{1}{2}\left((-2x + 3) - 3\right)$

3. Simplify: $g(f(x)) = -\frac{1}{2}\left(-2x + 3 - 3\right)$ $g(f(x)) = -\frac{1}{2}(-2x)$ $g(f(x)) = x$

Thus, $g(f(x)) = x$.

(c) Determine whether $f$ and $g$ are inverses

To determine if $f$ and $g$ are inverses of each other, both $f(g(x))$ and $g(f(x))$ must simplify to $x$. From parts (a) and (b), we found:

• $f(g(x)) = x$
• $g(f(x)) = x$

Since both conditions are satisfied, $f$ and $g$ are inverses of each other.

Final Answers:

(a) $f(g(x)) = x$
(b) $g(f(x)) = x$
(c) $f$ and $g$ are inverses of each other.

Would you like further details on any part of this process? Here are some related questions to explore further:

1. How do we verify if two functions are inverses graphically?
2. Can you explain why inverse functions reflect across the line $y = x$?
3. What is the domain and range of $f(x)$ and $g(x)$?
4. How would $f$ and $g$ behave if they were not inverses of each other?
5. How do we compute the inverse of a function algebraically?

Tip: Always check $f(g(x))$ and $g(f(x))$ when verifying if functions are inverses—it ensures accuracy!