91Ó°ÊÓ

The function \(f\) is periodic with period \(\pi\) and $$ \begin{gathered} f(x)=\sin x \quad \text { for } \quad 0 \leqslant x \leqslant \pi / 2 \\ f(x)=4\left(\pi^{2}-x^{2}\right) /\left(3 \pi^{2}\right) \quad \text { for } \quad \pi / 2

The function \(f(x)\) is defined for \(0 \leqslant x \leqslant 2\) by $$ \begin{array}{lll} f(x)=x & \text { for } & 0 \leqslant x \leqslant 1 \\ f(x)=(2-x)^{2} & \text { for } & 1

The function \(f\) is defined by $$ \begin{array}{lll} f(x)=\sin x & \text { for } & x \leqslant 0 \\ f(x)=x & \text { for } & x>0 \end{array} $$ Sketch the graphs of \(f(x)\) and its derivative \(f^{\prime}(x)\) for \(-\pi / 2

The function \(f\) is periodic with period 3 and $$ \begin{array}{llr} \left.f(x)=\sqrt{(} 9-4 x^{2}\right) & \text { for } & 0

Given that \(y=\frac{4 x^{2}+2 x+1}{x^{2}-x+1}\), determine the range of values taken by \(y\) for real values of \(x\).

Sketch the curve whose equation is \(y=1-\frac{1}{x+2}\).

Draw a sketch-graph of the curve whose equation is $$ y=x^{2}(2-x) $$ Hence, or otherwise, draw a sketch-graph of the curve whose equation is $$ y=\frac{1}{x^{2}(2-x)} $$ indicating briefly how the form of the curve has been derived.

Find the stationary points on the graphs of (a) \(y=\frac{x-1}{x^{2}}\) (b) \(y=\frac{1}{x}-1+\ln x\) Sketch the graphs of these functions.

\(R\) is the set of all real numbers. The mapping \(g\) is defined by $$ g: x \rightarrow \frac{2 x+1}{x-1}, \quad(x \in R, \quad x \neq 1) $$ State the range of \(g\) and sketch the graph of \(y=g(x)\). Define the mapping \(g^{-1}\).

Find the range of values of \(k\) for which the function $$ \frac{x^{2}-1}{(x-2)(x+k)} $$ where \(x\) is real, takes all real values.