The function $\mathrm{f}$ is defined by $$\mathrm{f}(x)=x^2+x+3,\ x\geq a$$ where $a$ is a

Question

The function $\mathrm{f}$ is defined by $$\mathrm{f}(x)=x^2+x+3,\ x\geq a$$ where $a$ is a real constant. State the least value of $a$ for which $\mathrm{f}^{-1}$ exists.[1] Using this value of $a$, find $\mathrm{f}^{-1}(x)$.[4] Find the domain of $\mathrm{f}^{-1}$.[2] Sketch, on the same axes, the graphs of $\mathrm{f}$ and $\mathrm{f}^{-1}$, labelling clearly the coordinates of any endpoints.[3] The function $\mathrm{g}$ is defined by $\mathrm{g}(x)=x^2+x+3$. Find two possible linear functions $\mathrm{h}$ such that $(\mathrm{g}\circ\mathrm{h})(x)=4x^2-14x+15$.[6]

Accepted Answer

(a) $a=-\frac12$. (b) $\mathrm{f}^{-1}(x)=\frac{-1+\sqrt{4x-11}}{2}$. (c) $x\geq\frac{11}{4}$. (d) Endpoints $\left(-\frac12,\frac{11}{4}\right)$ and $\left(\frac{11}{4},-\frac12\right)$; the curves are reflections in $y=x$. (e) $\mathrm{h}(x)=2x-4$ or $\mathrm{h}(x)=-2x+3$.

IB AA HL Q9: Quadratic inverse and composition (paper unspecified)

IB AA HL Q9: Quadratic inverse and composition (paper unspecified)

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