The function $\mathrm{f}$ is defined by \[\mathrm{f}: x \mapsto x^2 - 4x + 5, x \in \mathbb{R}, x

Question

The function $\mathrm{f}$ is defined by $$\mathrm{f}: x \mapsto x^2 - 4x + 5, x \in \mathbb{R}, x > 2.$$ Find $\mathrm{f}^{-1}(x)$ and write down the domain of $\mathrm{f}^{-1}$.[3] Sketch the graph of $\mathrm{f}$ and $\mathrm{f}^{-1}$ on the same diagram.[2] Explain why the $x$-coordinate of the point of intersection of the curves in part (b) satisfies the equation $$x^2 - 5x + 5 = 0$$ and find the exact value of this $x$-coordinate.[3]

Accepted Answer

(a) $\mathrm f^{-1}(x)=2+\sqrt{x-1}$, domain $(1,\infty)$ (b) See graph. (c) $x=\dfrac{5+\sqrt5}{2}$

Functions - Sketching a Function and Its Inverse

Functions - Sketching a Function and Its Inverse

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