1. **State the problem:** We are given three functions: $$f(x) = 3 - 4x, \quad g(x) = \frac{1}{3}x - 2, \quad h(x) = \frac{2x - 3}{5x - 7}$$ We need to solve the following: i. Find the value of $f(-3) + g(6)$. ii. Calculate the value of $x$ for which $f(x) = 5$. iii. Determine a simplified expression for $h^{-1}(x)$ (the inverse of $h$). iv. Determine a simplified expression for the composite function $h(g(x))$. --- 2. **Step i: Find $f(-3) + g(6)$** Calculate $f(-3)$: $$f(-3) = 3 - 4(-3) = 3 + 12 = 15$$ Calculate $g(6)$: $$g(6) = \frac{1}{3} \times 6 - 2 = 2 - 2 = 0$$ Sum: $$f(-3) + g(6) = 15 + 0 = 15$$ --- 3. **Step ii: Calculate $x$ such that $f(x) = 5$** Start with the equation: $$3 - 4x = 5$$ Subtract 3 from both sides: $$3 - 4x - 3 = 5 - 3 \implies -4x = 2$$ Divide both sides by $-4$: $$x = \frac{2}{\cancel{-4}} \times \cancel{-1} = -\frac{1}{2}$$ So, $$x = -\frac{1}{2}$$ --- 4. **Step iii: Find the inverse function $h^{-1}(x)$** Start with: $$y = \frac{2x - 3}{5x - 7}$$ Swap $x$ and $y$ to find the inverse: $$x = \frac{2y - 3}{5y - 7}$$ Multiply both sides by denominator: $$x(5y - 7) = 2y - 3$$ Distribute: $$5xy - 7x = 2y - 3$$ Group $y$ terms on one side: $$5xy - 2y = 7x - 3$$ Factor out $y$: $$y(5x - 2) = 7x - 3$$ Divide both sides by $(5x - 2)$: $$y = \frac{7x - 3}{5x - 2}$$ Therefore, $$h^{-1}(x) = \frac{7x - 3}{5x - 2}$$ --- 5. **Step iv: Find the composite function $h(g(x))$** Recall: $$g(x) = \frac{1}{3}x - 2$$ Substitute $g(x)$ into $h$: $$h(g(x)) = \frac{2\left(\frac{1}{3}x - 2\right) - 3}{5\left(\frac{1}{3}x - 2\right) - 7}$$ Simplify numerator: $$2 \times \frac{1}{3}x - 2 \times 2 - 3 = \frac{2}{3}x - 4 - 3 = \frac{2}{3}x - 7$$ Simplify denominator: $$5 \times \frac{1}{3}x - 5 \times 2 - 7 = \frac{5}{3}x - 10 - 7 = \frac{5}{3}x - 17$$ So, $$h(g(x)) = \frac{\frac{2}{3}x - 7}{\frac{5}{3}x - 17}$$ Multiply numerator and denominator by 3 to clear fractions: $$h(g(x)) = \frac{2x - 21}{5x - 51}$$ --- **Final answers:** $$\boxed{\begin{cases} f(-3) + g(6) = 15 \\ x = -\frac{1}{2} \text{ such that } f(x) = 5 \\ h^{-1}(x) = \frac{7x - 3}{5x - 2} \\ h(g(x)) = \frac{2x - 21}{5x - 51} \end{cases}}$$