Question 3.33: Sketch the graph of the function f : R → R given by f(x) = ......
Sketch the graph of the function f\colon\mathbb{R}\rightarrow\mathbb{R} given by
f(x) = 3 + 2x – x².
Show that f is not injective. Determine Im f and find a subset A of \mathbb{R} such that the restriction of f to A induces a bijection g : A → Im f Obtain a formula for the inverse of this bijection.
The graph of/is shown in Fig. S3.40.
Since, for example, f(3) =f(-1) = 0 we see that f is not injective.
\operatorname{Im}\ f=\{x∈ {\mathbb{R}}|\ x\leqslant4\rangle. Consider A\ =\{x\in\mathbb{R}\mid x\geqslant1\}. Clearly, the restriction of f to A is a bijection from A to Im f. To determine the inverse of this bijection we must find the solution of y = 3 + 2x — x² which lies in A. The solutions of this quadratic equation are x=1\pm\sqrt{(4-y)} of which only \displaystyle1\,+\,\sqrt{(4-y)} lies in A. Hence the required inverse is described by x\rightarrow1+\sqrt{(4-x)}.
