Functions (AQA GCSE Further Maths): Exam Questions

A function f is given by

$\begin{array}{rcl}\mathrm{f}\left(x\right)& =& 4xx<0\\ & =& x^2–8x0⩽x⩽8\\ & =& 16–2xx>8\end{array}$

A sketch of $y=\mathrm{f}\left(x\right)$is shown.

Work out all the values of $x$ for which  $\mathrm{f}\left(x\right)=-12$.

A function is given by 

  $\begin{array}{rcl}\mathrm{f}\left(x\right)& =& -2x-1⩽x<0\\ & =& x(4–x)0⩽x<3\\ & =& 2x–33⩽x⩽4\end{array}$

Draw the graph of $y=\mathrm{f}\left(x\right)$ on the grid.

$\mathrm{f}\left(x\right)=\frac{x-3}{2x}$

Solve $\mathrm{f}(x+1)–\mathrm{f}\left(2x\right)=0.5$

You must show your working.

$\mathrm{g}\left(x\right)=5-x^2$

The domain of $\mathrm{g}\left(x\right)$ is $–2⩽x⩽1$

Work out the range of $\mathrm{g}\left(x\right)$.

$\mathrm{f}\left(x\right)=x^2-7$     for all values of $x$

$\mathrm{g}\left(x\right)=1–3x$     for $–4⩽x⩽4$

Work out the range of $\mathrm{f}\left(x\right)$.

Give your answer as an inequality.

Work out the range of $\mathrm{g}\left(x\right)$.

Give your answer as an inequality.

Solve $2\mathrm{f}\left(x\right)$ =g$\left(x\right)$

You must show your working.

Give your answers to 3 decimal places.

$\mathrm{f}\left(x\right)=\frac{x}{2x+1}$  for positive values of $x$.

Work out   $\mathrm{f}\left(x+1\right)-\mathrm{f}\left(x\right)$

Give your answer as a fraction in its simplest form. You must show your working.

The function $\mathrm{f}$ is given by    $\mathrm{f}\left(x\right)=3x–5$

The range is $13<\mathrm{f}\left(x\right)<19$

Work out the domain of the function.

The function $\mathrm{g}$ is given by $\mathrm{g}\left(x\right)=x^2–4$ with domain $–1<x<3$

Work out the range of the function.

The function h is given by $\mathrm{h}\left(x\right)=\frac{3+x}{2}$

Work out${\mathrm{h}}^{-1}\left(x\right)$.

$\mathrm{f}\left(x\right)={\left(\frac{9x}{2}\right)}^{-1}$

$\mathrm{g}\left(x\right)=\sqrt{1-p{x}^3}$ where $p$ is a constant.

Given that $\mathrm{f}\left(\frac{1}{3}\right)=\mathrm{g}\left(\frac{1}{3}\right)$ work out the value of $p$.

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

$\mathrm{g}$ is a function such that  $\mathrm{gf}\left(x\right)={(x+2)}^{12}$

Work out an expression for $\mathrm{g}\left(x\right)$.

$\mathrm{h}\left(x\right)=x^2+5$

$\mathrm{k}$ is a function such that $\mathrm{hk}\left(x\right)=4x^2+5$

Work out an expression for  $\mathrm{kh}\left(x\right)$.

$\mathrm{f}\left(x\right)$ is a function with domain all values of $x$.

$\mathrm{f}\left(x\right)=\sqrt{x^2+6x-a}$  where $a$is a constant.

Work out the possible values of $a$. Give your answer as an inequality.

For each of these two function machines, when the input is $a$ the output is $b$.

$k>0$    and    $k\ne 1$    and     $a>0$

Work out an expression for $a$ in terms of $k$.

Give your answer in its simplest form.