1. **State the problem:** We have three functions: $$ f(x) = \frac{4x+5}{1-2x}, \quad g(x) = 3x + 5, \quad h(x) = px^2 + q - 1 $$ We need to find values and expressions as specified. 2. **(a)(i) Find $f\left(-\frac{1}{2}\right)$:** Substitute $x = -\frac{1}{2}$ into $f(x)$: $$ f\left(-\frac{1}{2}\right) = \frac{4\times \left(-\frac{1}{2}\right)+5}{1 - 2\times \left(-\frac{1}{2}\right)} = \frac{-2 +5}{1 +1} = \frac{3}{2} = 1.5 $$ 3. **(a)(ii) Find $g(-2)$:** Substitute $x = -2$ into $g(x)$: $$ g(-2) = 3 \times (-2) + 5 = -6 + 5 = -1 $$ 4. **(a)(iii) Find $q$ given $h(0) = -4$:** Substitute $x=0$ in $h(x)$: $$ h(0) = p \times 0^2 + q - 1 = q - 1 $$ Given $h(0) = -4$: $$ q - 1 = -4 \implies q = -3 $$ 5. **(a)(iv) Find $x$ that makes $f$ undefined:** $f$ is undefined when denominator is zero: $$ 1 - 2x = 0 \implies 2x = 1 \implies x = \frac{1}{2} $$ 6. **(b)(i) Find expression for $f^{-1}(x)$:** Set $y = f(x) = \frac{4x+5}{1-2x}$. To find $f^{-1}(x)$, solve for $x$ in terms of $y$: $$ y = \frac{4x+5}{1-2x} $$ Multiply both sides by $(1-2x)$: $$ y(1 - 2x) = 4x + 5 $$ $$ y - 2xy = 4x + 5 $$ Rearrange terms: $$ y - 5 = 4x + 2xy = x(4 + 2y) $$ Thus, $$ x = \frac{y - 5}{4 + 2y} $$ Replace $y$ by $x$ for inverse function: $$ f^{-1}(x) = \frac{x - 5}{4 + 2x} $$ 7. **(b)(ii) Find expression for $fg(x)$:** By definition, $$ fg(x) = f(g(x)) = f(3x + 5) $$ Substitute into $f$: $$ f(3x + 5) = \frac{4(3x + 5) + 5}{1 - 2(3x + 5)} = \frac{12x + 20 + 5}{1 - 6x - 10} = \frac{12x + 25}{-6x - 9} $$ 8. **(c) Find $x$ such that $f^{-1}(x) = -1$:** From (6), $$ f^{-1}(x) = \frac{x - 5}{4 + 2x} = -1 $$ Multiply both sides by denominator: $$ x - 5 = -1 (4 + 2x) = -4 - 2x $$ $$ x + 2x = -4 + 5 $$ $$ 3x = 1 $$ $$ x = \frac{1}{3} $$ **Final answers:** - (a)(i) $f\left(-\frac{1}{2}\right) = \frac{3}{2}$ - (a)(ii) $g(-2) = -1$ - (a)(iii) $q = -3$ - (a)(iv) $x = \frac{1}{2}$ makes $f$ undefined - (b)(i) $f^{-1}(x) = \frac{x - 5}{4 + 2x}$ - (b)(ii) $fg(x) = \frac{12x + 25}{-6x - 9}$ - (c) $x = \frac{1}{3}$ where $f^{-1}(x) = -1$