Functions (AQA A Level Maths: Pure): Exam Questions

The function $\mathrm{f}$ is defined as

$\mathrm{f}:x\mapsto \frac{x^2+1}{x^2}x\in \mathrm{ℝ},x\ne 0$

 Show that $\mathrm{f}$ can be written in the form

 $\mathrm{f}:x\mapsto a+\frac{b}{x^2}x\in \mathrm{ℝ},x\ne 0$

where $a$ and $b$ are constants to be found.

Explain why the inverse of $\mathrm{f}$ does not exist.

The domain of $\mathrm{f}$ is changed to $x>0$.

 Find ${\mathrm{f}}^{-1}\left(x\right)$ and state its domain and range.

State whether the following mappings are one-to-one or many-to-one:

(i)       $\mathrm{f}:x\mapsto 2-x^3$

(ii) $\mathrm{f}:x\mapsto \sin x$

(iii)     $\mathrm{f}:x\mapsto \frac{1}{x^2}$

(iv) $\mathrm{f}:x\mapsto \ln x$

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$\mathrm{f}\left(x\right)=2x^2+5,x\in \mathrm{ℝ},x>0$

$\mathrm{g}\left(x\right)=3x-1,x\in \mathrm{ℝ}$

Find $\mathrm{fg}\left(2\right)$.

Find ${\mathrm{f}}^{-1}\left(x\right)$ and state its domain.

The functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ are defined as follows:

$\mathrm{f}\left(x\right)=\frac{1}{2}(4x-3)x\in \mathrm{ℝ}$

$\mathrm{g}\left(x\right)=0.5x+0.75x\in \mathrm{ℝ}$

Find and simplify

(i) $\mathrm{fg}\left(x\right)$

(ii) $\mathrm{gf}\left(x\right)$

Explain why ${\mathrm{f}}^{-1}\left(x\right)=\mathrm{g}\left(x\right)$ and state the domain and range of ${\mathrm{f}}^{-1}\left(x\right)$.

The function $\mathrm{f}\left(x\right)$ is defined as

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

Sketch the curve $y=\mathrm{f}\left(x\right)$, labelling the coordinates of any points where the curve meets the coordinate axes.

Find and label the coordinates of the turning point on your sketch.

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

The functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ are defined as follows:

$\mathrm{f}\left(x\right)=|x-9|x\in \mathrm{ℝ}$

$\mathrm{g}\left(x\right)=x^{2}x\in \mathrm{ℝ}$

Sketch the graph of $y=\mathrm{fg}\left(x\right)$ and label the coordinates of any points where the graph meets the coordinate axes.

Find the number of solutions to the equation

$\mathrm{fg}\left(x\right)=k$

in the cases when:

(i) $k=5$

(ii) $k=9$

Solve

$\mathrm{fg}\left(x\right)=0$

Find the largest possible domains of the following functions:

(i) $\mathrm{f}:x\mapsto \sqrt{x}$

(ii) $\mathrm{f}:x\mapsto \ln (x-2)$

(iii) $\mathrm{f}:x\mapsto \arcsin x$

On the same axes, sketch the graphs of $y=\mathrm{f}\left(x\right)$ and $y=\left|\mathrm{g}\left(x\right)\right|$ where

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

 $\mathrm{g}\left(x\right)=2x+4x\in \mathrm{ℝ}$

Label the points at which the graphs meet the coordinate axes.

Solve the equation 

${\left(x+2\right)}^2=|2x+4|$

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by:

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

 $\mathrm{g}\left(x\right)=1-3xx\in \mathrm{ℝ}$

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

Find expressions for

(i)  $\mathrm{fg}\left(x\right)$

(ii)  $\mathrm{gf}\left(x\right)$

Give your answers in the form $a{x}^2+bx+c$ where $a$, $b$ and $c$ are constants to be found.

Solve the equation

$\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)+1$

The graph of $y=\mathrm{f}\left(x\right)$ is shown below.

The dotted line has the equation $y=x$

On the diagram, sketch the graph of $y={\mathrm{f}}^{-1}\left(x\right)$.

On the same axes, sketch the graphs of $y=\left|\mathrm{f}\left(x\right)\right|$  and $y=\left|\mathrm{g}\left(x\right)\right|$ where

$\mathrm{f}\left(x\right)=3x-1x\in \mathrm{ℝ}$

$\mathrm{g}\left(x\right)=2x+2x\in \mathrm{ℝ}$

Label the coordinates of the points at which the graphs meet the coordinate axes.

Solve the equation

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

Which of the solutions to $\left|\mathrm{f}\left(x\right)\right|=\left|\mathrm{g}\left(x\right)\right|$ is also a solution to $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$?

Solve the equation $|x^2-4|=3$ , giving your answers in exact form.

The functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ are defined by

$\mathrm{f}\left(x\right)={\mathrm{e}}^{x-2}x\in \mathrm{ℝ}$

 $\mathrm{g}\left(x\right)=2+\ln xx>0$

Find and simplify expressions for

(i) $\mathrm{fg}\left(x\right)$

(ii) $\mathrm{gf}\left(x\right)$

Explain why ${\mathrm{f}}^{-1}\left(x\right)=\mathrm{g}\left(x\right)$ and state the domain and range of ${\mathrm{f}}^{-1}\left(x\right)$.

Describe the transformation that would map the graph of $y=\mathrm{f}\left(x\right)$ on to the graph of $y=\mathrm{g}\left(x\right)$.

The functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$ are defined by

$\mathrm{f}\left(x\right)=|x-2|-5x\in \mathrm{ℝ}$

$\mathrm{g}\left(x\right)=\left|x\right|x\in \mathrm{ℝ}$

Sketch the graph of $y=\mathrm{gf}\left(x\right)$.

Label clearly the coordinates of any points where the graph meets the coordinate axes.

Find the number of solutions to the equation

$\mathrm{gf}\left(x\right)=k$

in the cases when:

(i) $k=1$

(ii) $k=10$

Solve the equation

$\mathrm{gf}\left(x\right)=2$

Sketch the curve with equation $y=2\left|x-3\right|-4$. State clearly the coordinates of the minimum point and the points where the curve crosses the axes.

Solve the inequality $2\left|x-3\right|-4<\frac{1}{2}x+1$.

Write down the set of values of $k$ for which the equation $2\left|x-3\right|-4=k$ has exactly two distinct real solutions.

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

$\mathrm{f}\left(x\right)=x^2-2x\in \mathrm{ℝ}$

$\mathrm{g}\left(x\right)=1-\frac{2}{x}x\in \mathrm{ℝ},x\ne 0$

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

Leaving your answers as single algebraic fractions, find expressions for

(i)   $\mathrm{fg}\left(x\right)$

(ii) $\mathrm{gf}\left(x\right)$

Use algebra to solve the equation

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

On the same axes, sketch the graphs of $y=\mathrm{f}\left(x\right)$ and $y=\left|\mathrm{g}\left(x\right)\right|$ where

$\mathrm{f}\left(x\right)=\sqrt{x}x\ge 0$

$\mathrm{g}\left(x\right)=2x-3x\in \mathrm{ℝ}$

Label the points at which the graphs meet the coordinate axes.

Solve the equation

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

Which (if any) of the solutions to $\mathrm{f}\left(x\right)=\left|\mathrm{g}\left(x\right)\right|$ are not solutions to $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$?

A function is defined by

$\mathrm{f}\left(x\right)=a|x+p|+qx\in \mathrm{ℝ}$

where $a,p,q\in \mathrm{ℝ}$.

The graph of $y=\mathrm{f}\left(x\right)$is shown below, where

• $A\left(3,5\right)$ is the local maximum point

• $\left(0,-7\right)$ is the $y$-intercept

Find the values of $a,p$ and $q.$

Solve

$\mathrm{f}\left(x\right)=\left|7-2x\right|$

Solve the equation 

$|x^2-9|=6-0.25x^2$

giving your answers in exact form.

The function $\mathrm{f}$ is defined by

$\mathrm{f}\left(x\right)=(x-3{)}^2(x-4{)}^{2}2\le x\le 5$

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

You must show your working clearly.

The domain of $\mathrm{f}$ is changed to $x\le 2$

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

Find a different domain of $\mathrm{f}$ that has the same range as in part (b).

The graph of $y=\mathrm{f}\left(x\right)$ is shown in the diagram below. The dotted line has equation $y=x$.

You are given that

• $y=\mathrm{f}\left(x\right)$ has rotational symmetry about the origin

• For $x>0$, the vertices of the graph have coordinates $\left(3,6\right)$, $\left(6,6\right)$ and $\left(9,0\right)$

• $y=\mathrm{f}\left(x\right)$ does not continue for $x^2>81$

(i) Find the domain and range of the function $\mathrm{f}$.

(ii) On the diagram above, sketch the reflection of $y=\mathrm{f}\left(x\right)$ in the line $y=x$ and explain why this cannot be the graph of $y={\mathrm{f}}^{-1}\left(x\right)$

(i) Find the largest restricted domain of $\mathrm{f}$ that includes the value $x=0$ such that ${\mathrm{f}}^{-1}\left(x\right)$ exists.

(ii) Assuming the domain of $\mathrm{f}$ in part (b)(ii) is used, find the domain and range of the function ${\mathrm{f}}^{-1}$.

The function $\mathrm{f}$ is defined by

$\mathrm{f}:x\mapsto \sqrt{25-x^2}-5\le x\le 5$

Determine whether or not the inverse of $\mathrm{f}$ exists.

The domain is changed such that

• the inverse of $\mathrm{f}$ exists

• the graph of $y=\mathrm{f}\left(x\right)$ lies in the first quadrant only

• the domain of $\mathrm{f}$ is as large as possible

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

The domain of $\mathrm{f}$ is changed to $-5\le x\le 0$.

Find an expression for ${\mathrm{f}}^{-1}\left(x\right)$.

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

The functions $\mathrm{f}$ and $\mathrm{g}$ are defined by

 $\mathrm{f}\left(x\right)=(x-1{)}^2-4x\ge 1$

$\mathrm{g}\left(x\right)=1+\sqrt{x+4}x\ge -4$

Find and simplify expressions for

(i)  $\mathrm{fg}\left(x\right)$

(ii)  $\mathrm{gf}\left(x\right)$

Write down ${\mathrm{f}}^{-1}\left(x\right)$ and state its domain and range.

Describe the transformation that would map the graph of $y=\mathrm{f}\left(x\right)$ on to the graph of $y=\mathrm{g}\left(x\right)$.

Find the coordinates of the point of intersection between the curves  $y=\mathrm{f}\left(x\right)$ and $y=\mathrm{g}\left(x\right)$.