State the coordinates of the turning points of the graph of $y = x^3 - 3x^2 - 9x + 5$.[1] The func

Question

State the coordinates of the turning points of the graph of $y = x^3 - 3x^2 - 9x + 5$.[1] The function $\mathrm{f}$ is defined by$$\mathrm{f}(x) = x^3 - 3x^2 - 9x + 5, x \in \mathbb{R}, x \leq a$$ where $a$ is a constant. Find the largest possible value of $a$ such that $\mathrm{f}^{-1}$ exists, and state the domain of $\mathrm{f}^{-1}$ in this case.[2] The function $\mathrm{g}$ is defined by$$\mathrm{g}(x) = \frac{1 - 2x}{x + 2}, x \in \mathbb{R}, x \neq -2.$$ Find $\mathrm{g}^2(x)$.[2] Hence find the possible expressions of $\mathrm{g}^n(x)$, where $n$ is a positive integer.[2] The function $\mathrm{h}$ is defined by$$\mathrm{h}(x) = x^2, x \in \mathbb{R}, x \leq 0.$$ Explain why the composite function $\mathrm{g}\mathrm{h}$ exists.[2] Find the range of $\mathrm{g}\mathrm{h}$.[2]

Accepted Answer

(a)(i) $(-1,10)$, $(3,-22)$. (ii) $a=-1$, $D_{\mathrm f^{-1}}=(-\infty,10]$. (b)(i) $\mathrm g^2(x)=x$, $x\ne-2$. (ii) Even $n$: $x$; odd $n$: $\frac{1-2x}{x+2}$, with $x\ne-2$ in both cases. (c)(i) $[0,\infty)\subset D_{\mathrm g}$, so the composite exists. (ii) $(-2,\frac{1}{2}]$.

2025 EJC P1 Q9

2025 EJC P1 Q9

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