CIE Additional Mathematics (0606)

CIE Additional Mathematics (0606)

Function

(1) 2018/m/22/q10

(a) The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\sqrt{1+x^{2}}$, for all real values of $x .$ The graph of $y=\mathrm{f}(x)$ is given below.

(i) Explain, with reference to the graph, why $f$ does not have an inverse.    [1]

(ii) Find $\mathrm{f}^{2}(x)$.    [2]

(b) The function $g$ is defined, for $x>k$, by $g(x)=\sqrt{1+x^{2}}$ and $g$ has an inverse.

(i) Write down a possible value for $k$.    [1]

(ii) Find $\mathrm{g}^{-1}(x)$.    [2]

(c) The function $\mathrm{h}$ is defined, for all real values of $x$, by $h(x)=4 \mathrm{e}^{x}+2$. Sketch the graph of $y=\mathrm{h}(x)$. Hence, on the same axes, sketch the graph of $y=\mathrm{h}^{-1}(x)$. Give the coordinates of any points where your graphs meet the coordinate axes.    [4]

Short Answer

(a)(i) not $1-1$

(ii) $\sqrt{2+x^{2}}$

(b)(i) $\geq 0$

(ii) $\sqrt{x^{2}-1}$

(iii) (0,6), (6,0)

(2) 2018/s/21/q5

The function $\mathrm{f}$ is defined by $\mathrm{f}(x)=\frac{1}{2 x-5}$ for $x>2.5$.

(i) Find an expression for $\mathrm{f}^{-1}(x)$.                                                         [2]

(ii) State the domain of $\mathrm{f}^{-1}(x)$.                                                             [1]

(iii) Find an expression for $\mathrm{f}^{2}(x)$, giving your answer in the form $\frac{a x+b}{cx+d}$, where $a, b, c$ and $d$ are integers to be found.                    [3]

Short Answer
(i) $f^{-1}(x)=\frac 12[\frac1x+5]$
(ii) $x>0$
(iii) $\frac{2 x-5}{-10 x+27}$

3 2018/s/22/q10

(a) (i) On the axes below, sketch the graph of $y=|(x+3)(x-5)|$ showing the coordinates of the points where the curve meets the $x$ -axis.                                                         [2]

(ii) Write down a suitable domain for the function $\mathrm{f}(x)=|(x+3)(x-5)|$ such that $\mathrm{f}$ has an inverse.                                                                                 [1]

(b) The functions $g$ and $h$ are defined by

                        $g(x)=3x-1$ for $x>1$

                        $h(x)=\frac{4}{x}$ for $x\neq 0$

(i) Find hg $(x)$.                                                                                                            [1]

(ii) Find (hg) $^{-1}(x)$.                                                                                               [2]

(c) Given that $\mathrm{p}(a)=b$ and that the function $\mathrm{p}$ has an inverse, write down $\mathrm{p}^{-1}(b)$.                                                                              [1]

Short Answer
(a)(ii) $x \geq 5$ or $x \leq-3$ or $-3 \leq x \leq 1$ or $1 \leq x \leq 5$
(bi) $\frac{4}{3 x-1}$
(bii) $\frac{(4+x) }{(3 x)}$
(c) a