1. **State the problem:** We are given two functions: $$h(x) = \frac{10}{x - 3}$$ and $$g(x) = 3x - 2$$ We need to evaluate some function values and find expressions for inverse and composite functions. 2. **Evaluate (a)(i) $g(4)$:** Substitute $x=4$ into $g(x)$: $$g(4) = 3(4) - 2 = 12 - 2 = 10$$ 3. **Evaluate (a)(ii) $h(g(4))$:** First, find $g(4)$ from step 2, which is 10. Now substitute into $h(x)$: $$h(g(4)) = h(10) = \frac{10}{10 - 3} = \frac{10}{7}$$ 4. **Find (b)(i) the inverse function $h^{-1}(x)$:** Start with: $$y = \frac{10}{x - 3}$$ Swap $x$ and $y$ to find inverse: $$x = \frac{10}{y - 3}$$ Multiply both sides by $y - 3$: $$x(y - 3) = 10$$ Distribute: $$xy - 3x = 10$$ Add $3x$ to both sides: $$xy = 10 + 3x$$ Divide both sides by $x$: $$y = \frac{10 + 3x}{x}$$ This is the inverse function: $$h^{-1}(x) = \frac{10 + 3x}{x}$$ 5. **Find (b)(ii) the composite function $gg(x)$:** This means $g(g(x))$. Start with: $$g(x) = 3x - 2$$ Substitute $g(x)$ into $g$: $$g(g(x)) = g(3x - 2) = 3(3x - 2) - 2 = 9x - 6 - 2 = 9x - 8$$ **Final answers:** - $g(4) = 10$ - $h(g(4)) = \frac{10}{7}$ - $h^{-1}(x) = \frac{10 + 3x}{x}$ - $gg(x) = 9x - 8$