The problem asks us to algebraically determine whether the given pairs of functions are inverses. To verify, we must check if $f(g(x)) = x$ and $g(f(x)) = x$ for each pair of functions. Let’s go through each case step-by-step.

Part (a):

$f(x) = -2x + 2 \quad \text{and} \quad g(x) = \frac{1}{2}x + 2$

1. Check $f(g(x))$: Substitute $g(x) = \frac{1}{2}x + 2$ into $f(x)$: $f(g(x)) = f\left(\frac{1}{2}x + 2\right) = -2\left(\frac{1}{2}x + 2\right) + 2$ Simplify: $f(g(x)) = -x - 4 + 2 = -x - 2$ Since $f(g(x)) \neq x$, $f$ and $g$ are not inverses.

Part (b):

$f(x) = 2x^3 + 4 \quad \text{and} \quad g(x) = \sqrt[3]{\frac{x - 3}{2}}$

1. Check $f(g(x))$: Substitute $g(x) = \sqrt[3]{\frac{x - 3}{2}}$ into $f(x)$: $f(g(x)) = 2\left(\sqrt[3]{\frac{x - 3}{2}}\right)^3 + 4$ Simplify: $f(g(x)) = 2\left(\frac{x - 3}{2}\right) + 4 = (x - 3) + 4 = x + 1$ Since $f(g(x)) \neq x$, $f$ and $g$ are not inverses.

Part (c):

$f(x) = \sqrt{3x - 2} \quad \text{and} \quad g(x) = \frac{x^2}{3} + \frac{2}{3}$

1. Check $f(g(x))$: Substitute $g(x) = \frac{x^2}{3} + \frac{2}{3}$ into $f(x)$: $f(g(x)) = \sqrt{3\left(\frac{x^2}{3} + \frac{2}{3}\right) - 2}$ Simplify: $f(g(x)) = \sqrt{x^2 + 2 - 2} = \sqrt{x^2}$ Since $\sqrt{x^2} = |x|$, $f(g(x)) \neq x$ for all $x$ (e.g., $x < 0$).
   Thus, $f$ and $g$ are not inverses.

Part (d):

$g(x) = -\frac{3}{4}x + 5 \quad \text{and} \quad h(x) = -\frac{4}{3}x - \frac{20}{3}$

1. Check $g(h(x))$: Substitute $h(x) = -\frac{4}{3}x - \frac{20}{3}$ into $g(x)$: $g(h(x)) = -\frac{3}{4}\left(-\frac{4}{3}x - \frac{20}{3}\right) + 5$ Simplify: $g(h(x)) = x + \frac{15}{3} - 5 = x$

2. Check $h(g(x))$: Substitute $g(x) = -\frac{3}{4}x + 5$ into $h(x)$: $h(g(x)) = -\frac{4}{3}\left(-\frac{3}{4}x + 5\right) - \frac{20}{3}$ Simplify: $h(g(x)) = x - \frac{20}{3} - \frac{20}{3} = x$

Since both $g(h(x)) = x$ and $h(g(x)) = x$, $g$ and $h$ are inverses.

Final Results:

• (a) Not inverses
• (b) Not inverses
• (c) Not inverses
• (d) Inverses

Would you like further clarification or a more detailed breakdown?

Here are 5 related questions you can explore:

1. How do you find the inverse of a linear function?
2. What are the conditions for two functions to be inverses?
3. Can a function have more than one inverse? Why or why not?
4. How do restrictions on the domain affect whether two functions are inverses?
5. How can you graphically verify if two functions are inverses?

Tip: Always verify both $f(g(x)) = x$ and $g(f(x)) = x$ to confirm inverses algebraically!