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

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(i) $\mathrm{f}:x\mapsto x^2$

(ii) $\mathrm{f}:x\mapsto 3x+1$

(iii) $\mathrm{f}:x\mapsto {\left(x+1\right)}^3$

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

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

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

Sketch the graph of $y=\mathrm{f}\left(x\right)$, giving the coordinates of any points where the graph intercepts the coordinate axes and the coordinates of the turning point.

Write down the range of $\mathrm{f}\left(x\right)$.

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

$\mathrm{f}\left(x\right)=x^2—4x\ge 0$

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

If the domain of $\mathrm{f}\left(x\right)$ is changed to $x\le 0$, what is the range of $\mathrm{f}\left(x\right)$?

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

$$\begin{gathered} \mathrm{f}\left(x\right)=x^{2}x\in \mathrm{ℝ} \\ g\left(x\right)=4x—3x\in \mathrm{ℝ} \end{gathered}$$

Write down the range of $\mathrm{f}\left(x\right)$.

Find 

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

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

Solve the equation $\mathrm{f}\left(x\right)=g\left(x\right)$.

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

(i) Use the graph to write down the domain and range of $\mathrm{f}\left(x\right)$.

(ii) Given that the point (1, 1) lies on the dotted line, write down the equation of the line.

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

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

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

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

Solve the equation $\mathrm{f}\left(x\right)=\left|g\left(x\right)\right|.$

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

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

Show that $\mathrm{f}\left(x\right)$ can be written in the form

$\mathrm{f}:x\mapsto 1+\frac{1}{x^2}$

Explain why the inverse of $\mathrm{f}\left(x\right)$ does not exist and suggest an adaption to its domain so the inverse does exist.

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

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

Solve the equation $\left|4x+2\right|=5.$

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

$$\begin{gathered} \mathrm{f}\left(x\right)=\frac{1}{2}\left(4x-3\right)x\in \mathrm{ℝ} \\ g\left(x\right)=0.5x+0.75x\in \mathrm{ℝ} \end{gathered}$$

Find

(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.

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

(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) $\mathrm{f}:x\mapsto \mathrm{In}x$

It is given 

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

Write down the domain of the function $\text{f}\left(x\right)$.

Sketch the graph of $y=\mathrm{f}\left(x\right)$, stating the coordinates of any intersections with the coordinate axes and the equations of any asymptotes.

Write down the range of $\mathrm{f}\left(x\right)$.

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

$\mathrm{f}\left(x\right)=x{\left(x+3\right)}^2+1x\ge 0$

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

If the domain of $\mathrm{f}\left(x\right)$ is changed to $x\le 0$, what is the range of $\mathrm{f}\left(x\right)$?

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

$$\begin{gathered} \mathrm{f}\left(x\right)=3x^2+2x\in \mathrm{ℝ} \\ \mathrm{g}\left(x\right)=1—3xx\in \mathrm{ℝ} \end{gathered}$$

Write down the range of $\mathrm{f}\left(x\right)$.

Find

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

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

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.

(i) Use the graph to write down the domain and range of $\mathrm{f}\left(x\right)$.

(ii) Write down the equation of the dotted line on the graph.

On the diagram above 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

$$\begin{gathered} \mathrm{f}\left(x\right)=3x—1x\in \mathrm{ℝ} \\ g\left(x\right)=2x+2x\in \mathrm{ℝ} \end{gathered}$$

Label the points at which the graphs intersect 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 $\mathrm{f}\left(x\right)=\mathrm{g}\left(x\right)$?

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

$$\begin{gathered} \mathrm{f}\left(x\right)=e^{x-2}x\in \mathrm{ℝ} \\ g\left(x\right)=2+\mathrm{In}xx\in \mathrm{ℝ},x>0 \end{gathered}$$

Find

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

(ii) $\mathrm{gf}\left(x\right)$ 

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

The graphs of $\mathrm{f}\left(x\right)$ and ${\mathrm{f}}^{-1}\left(x\right)$ are drawn on the same axes.

Describe the transformation that would map one graph onto the other.

State whether the following mappings are one-to-one, many-to-one, one-to-many or many-to-many.

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

(ii) $\mathrm{f}:x\mapsto \left|\frac{1}{x}\right|$

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

(iv) $\mathrm{f}:x\mapsto \pm \sqrt{25-x^2}$

It is given

$\mathrm{f}\left(x\right)=4x^3+4x^2—7x+2$

Write down the domain and range of the function $\mathrm{f}\left(x\right)$.

Sketch the graph of $y=\mathrm{f}\left(x\right)$, stating the coordinates of any intersections with the coordinate axes. (You do not need to give the coordinates of any turning points.)

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

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

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

If the domain of $\mathrm{f}\left(x\right)$ is changed to $x\le 2$, what is the range of $\mathrm{f}\left(x\right)$?

State another domain for $\mathrm{f}\left(x\right)$ that would have the same effect as that in part (b).

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

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

Write down the range of $\mathrm{f}\left(x\right)$.

Leaving your answers as single fractions, find

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

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

The graphs of $y=\mathrm{f}\left(x\right)$ and $y=x$ (dotted line) are shown in the diagram below. $\mathrm{f}\left(x\right)$ has rotational symmetry about the origin and for $x>0$, there is a vertical line of symmetry at $x=4.5$.

(i) Use the graph to write down the domain and range of $\mathrm{f}\left(x\right)$.

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

(i) Given that the maximum solution to $\mathrm{f}\left(x\right)=6$ is $x=6$, state the restriction on the domain of $\mathrm{f}\left(x\right)$ such that ${\mathrm{f}}^{-1}\left(x\right)$ exists.

(ii) Hence, or otherwise, write down the domain and range of ${\mathrm{f}}^{-1}\left(x\right)$.

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

$$\begin{gathered} \mathrm{f}\left(x\right)=\sqrt{x}x\ge 0 \\ g\left(x\right)=2x—3x\in \mathrm{ℝ} \end{gathered}$$

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

Solve the equation $\mathrm{f}\left(x\right)=\left|g\left(x\right)\right|.$

Which of the solutions to $\mathrm{f}\left(x\right)=\left|g\left(x\right)\right|$ is not a solution to $\mathrm{f}\left(x\right)=g\left(x\right)$?

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

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

Explain why the inverse of $\mathrm{f}\left(x\right)$ does not exist.

Suggest an adaption to the domain of $\mathrm{f}\left(x\right)$ so the following conditions are met:

• the inverse of $\mathrm{f}\left(x\right)$exists,

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

• the domain of $\mathrm{f}\left(x\right)$ is as large as possible.

State the range for your adapted $\mathrm{f}\left(x\right)$.

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

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

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

$$\begin{gathered} \mathrm{f}\left(x\right)={\left(x-1\right)}^2-4x\in \mathrm{ℝ},x\ge 1 \\ g\left(x\right)=1+\sqrt{x+4}x\in \mathrm{ℝ},x\ge -4 \end{gathered}$$

Find

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

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

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

The graphs of $\mathrm{f}\left(x\right)$ and ${\mathrm{f}}^{-1}\left(x\right)$ are drawn on the same axes. Describe the transformation that would map one graph onto the other.

Find the coordinates of the point where the graphs of $y=\mathrm{f}\left(x\right)$ and $y={\mathrm{f}}^{-1}\left(x\right)$ meet.

The graph of the function $f\left(x\right)=a|x+p|+q$is shown below, where $a,p,q\in \mathrm{ℝ}$. The graph has a local maximum at the point A(3, 5)  and intersects the $y$-axis at (0,-7).

Find the values of $a,p$ and $q.$Hence solve $f\left(x\right)=0$.

Find the solutions to the equation $f\left(x\right)=\left|7-2x\right|$.