<ol class="abc-list"><li><div class="list-item-content"><span class="list-item-text"></span></div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">State the coordinates of the turning points of the graph of \(y = x^3 - 3x^2 - 9x + 5\).</span><span class="inline-marks">[1]</span></div></li><li><div class="list-item-paragraph">The function \(\mathrm{f}\) is defined by\[\mathrm{f}(x) = x^3 - 3x^2 - 9x + 5, x \in \mathbb{R}, x \leq a\]\nwhere \(a\) 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.<span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The function \(\mathrm{g}\) is defined by\[\mathrm{g}(x) = \frac{1 - 2x}{x + 2}, x \in \mathbb{R}, x \neq -2.\]</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find \(\mathrm{g}^2(x)\).</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Hence find the possible expressions of \(\mathrm{g}^n(x)\), where \(n\) is a positive integer.</span><span class="inline-marks">[2]</span></div></li></ol></li><li><div class="list-item-paragraph">The function \(\mathrm{h}\) is defined by\[\mathrm{h}(x) = x^2, x \in \mathbb{R}, x \leq 0.\]</div><ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Explain why the composite function \(\mathrm{g}\mathrm{h}\) exists.</span><span class="inline-marks">[2]</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the range of \(\mathrm{g}\mathrm{h}\).</span><span class="inline-marks">[2]</span></div></li></ol></li></ol>

State the coordinates of the turning points of the graph of \(y = x^3 - 3x^2 - 9x + 5\).

The function \(\mathrm{f}\) is defined by\n\[\mathrm{f}(x) = x^3 - 3x^2 - 9x + 5, x \in \mathbb{R}, x \leq a\]\nwhere \(a\) 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.

The function \(\mathrm{g}\) is defined by\n\[\mathrm{g}(x) = \frac{1 - 2x}{x + 2}, x \in \mathbb{R}, x \neq -2.\]

The function \(\mathrm{g}\) is defined by\[\mathrm{g}(x) = \frac{1 - 2x}{x + 2}, x \in \mathbb{R}, x \neq -2.\]

Hence find the possible expressions of \(\mathrm{g}^n(x)\), where \(n\) is a positive integer.

The function \(\mathrm{h}\) is defined by\n\[\mathrm{h}(x) = x^2, x \in \mathbb{R}, x \leq 0.\]

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.

Find the range of \(\mathrm{g}\mathrm{h}\).

Functions

International Baccalaureate Higher Level Mathematics

2025 EJC P1 Q9

2025 EJC P1 Q9