Functions (Cambridge (CIE) O Level Additional Maths): Exam Questions

It is given that $\mathrm{h}\left(x\right)=a+\frac{b}{x^2}$, where $a$ and $b$ are non-zero constants.

(i) Explain why $–2⩽x⩽2$ is not a suitable domain for $\mathrm{h}\left(x\right)$.

(ii) Given that $\mathrm{h}\left(1\right)=4$ and $\mathrm{h}\text{'}\left(1\right)=16$ , find the values of $a$ and $b$.

$\mathrm{f}:x\mapsto {(2x+3)}^2\mathrm{for}x>0$

Find the range of $\mathrm{f}$.

Explain why $\mathrm{f}$ has an inverse.

Find ${\mathrm{f}}^{-1}$.

State the domain of ${\mathrm{f}}^{-1}$.

Given that $\mathrm{g}:x\mapsto \ln (x+4)\mathrm{for}x>0$, find the exact solution of $\mathrm{fg}\left(x\right)=49$.

$\mathrm{g}\left(x\right)=x+5\mathrm{for}x\in \mathrm{ℝ}$
$\mathrm{h}\left(x\right)=\sqrt{2x-3}\mathrm{for}x\ge \frac{3}{2}$

Solve $\mathrm{gh}\left(x\right)=7$.

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined as follows.

$$\begin{gathered} \mathrm{f}\left(x\right)=x^2+4x\mathrm{for}x\in R \\ \mathrm{g}\left(x\right)=1+{\mathrm{e}}^{2x}\mathrm{for}x\in R \end{gathered}$$

Find the range of $\mathrm{f}$.

Write down the range of $\mathrm{g}$.

Find the exact solution of the equation $\mathrm{fg}\left(x\right)=21$, giving your answer as a single logarithm

$\mathrm{f}\left(x\right)=x^2+2x-3\mathrm{for}x\ge -1$

Given that the minimum value of $x^2+2x-3$ occurs when $x=-1$, explain why $\mathrm{f}\left(x\right)$ has an inverse.

On the axes below, sketch the graph of $y=\mathrm{f}\left(x\right)$ and the graph of $y={\mathrm{f}}^{-1}\left(x\right)$.
Label each graph and state the intercepts on the coordinate axes.

$\mathrm{f}\left(x\right)=3{\mathrm{e}}^{2x}+1$ for $x\in \mathrm{ℝ}$ $\mathrm{g}\left(x\right)=x+1$ for $x\in \mathrm{ℝ}$

(i) Write down the range of $\mathrm{f}$ and the range of $\mathrm{g}$.

(ii) Find${\mathrm{g}}^2\left(0\right)$.

(iii) Hence find ${\mathrm{fg}}^2\left(0\right)$ .

(iv) On the axes below, sketch the graphs of $y=\mathrm{f}\left(x\right)$ and $y={\mathrm{f}}^{–1}\left(x\right)$ . State the intercepts with the coordinate axes and the equations of any asymptotes.

The function$\mathrm{f}$ is defined by$\mathrm{f}\left(x\right)=\ln (2x+1)\mathrm{for}x\ge 0$.

Sketch the graph of $y=\mathrm{f}\left(x\right)$ and hence sketch the graph of $y={\mathrm{f}}^{-1}\left(x\right)$ on the axes below.

The function $\mathrm{g}$ is defined by $\mathrm{g}\left(x\right)={(x-4)}^2+1\mathrm{for}x\le 4$.

(i) Find an expression for ${\mathrm{g}}^{-1}\left(x\right)$ and state its domain and range.

[4]

(ii) Find and simplify an expression for $\mathrm{fg}\left(x\right)$.

[2]

(iii) Explain why the function $\mathrm{gf}$ does not exist.

[1]

$\mathrm{f}\left(x\right)=3+{\mathrm{e}}^x\mathrm{for}x\in \mathrm{ℝ}$
$\mathrm{g}\left(x\right)=9x-5\mathrm{for}x\in \mathrm{ℝ}$

Find the range of $\mathrm{f}$ and of $\mathrm{g}$.

Find the exact solution of ${\mathrm{f}}^{-1}\left(x\right)=\mathrm{g}\text{'}\left(x\right)$ .

Find the solution of ${\mathrm{g}}^2\left(x\right)=112$.