Modulus Functions (Cambridge (CIE) AS Maths: Pure 2): Exam Questions

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

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

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

$$\begin{gathered} \mathrm{f}\left(x\right)=\left|x-9\right|x\in \mathrm{ℝ} \\ \mathrm{g}\left(x\right)=x^{2}x\in \mathrm{ℝ} \end{gathered}$$

Sketch the graph of $y=\mathrm{fg}\left(x\right)$, stating the coordinates of all points where the graph intercepts the coordinate axes.

(i) How many solutions are there to the equation  $\mathrm{fg}\left(x\right)=5?$

(ii) How many solutions are there to the equation  $\mathrm{fg}\left(x\right)=9?$

Write down the solutions to the equation $\mathrm{fg}\left(x\right)=0.$

The turning point on the graph of $y=\mathrm{f}\left(x\right)$ has coordinates $(2,-5)$ as shown on the diagram below.

(i) On the diagram above sketch the graph of $y=\left|\mathrm{f}\left(\mathrm{x}\right)\right|+1$ and state the coordinates of the turning point. 

(ii) State the distance between the turning points on the graphs of $y=\mathrm{f}\left(x\right)$ and $y=\left|\mathrm{f}\left(\mathrm{x}\right)\right|+1$.

The diagram below shows the graph of $y=\mathrm{g}\left(x\right)$ where

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

Write down the equations of the two asymptotes.

Determine the equations of the two asymptotes on the graph of $y=\mathrm{g}\left(2x\right)-3$.

Determine the range of $\left|\mathrm{g}\left(3x\right)-2\right|$.

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

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

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

$\mathrm{f}:x\mapsto |3x-2|x\in \mathrm{ℝ}$

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 its inverse does exist, but also produces the maximum possible range for $\mathrm{f}\left(x\right)$.

Using your adaption from part (b), find an expression for  ${\mathrm{f}}^{-1}\left(x\right)$ and state its domain and range.

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

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

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

Sketch the graph of  $y=\mathrm{gf}\left(x\right)$, stating the coordinates of all points where the graph intercepts the coordinate axes.

(i) How many solutions are there to the equation  $\mathrm{gf}\left(x\right)=1?$ 

(ii) How many solutions are there to the equation  $\mathrm{gf}\left(x\right)=10?$

Solve the equation  $\mathrm{gf}\left(x\right)=2.$

The minimum point on the graph of $y=\mathrm{f}\left(x\right)$ has coordinates $(4,-8)$as shown on the diagram below.

Sketch the graph of $y=\left|\mathrm{f}\left(2\mathrm{x}\right)\right|-3$ and state the coordinates of the maximum point.

Find the exact distance between the minimum point on the graph of $y=\mathrm{f}\left(x\right)$ and the maximum point on the graph of $y=\left|\mathrm{f}\left(2\mathrm{x}\right)\right|-3.$

On the same axes sketch the graphs of $y=\mathrm{p}\left(x\right)$and $y={\mathrm{p}}^{-1}\left(x\right)$, where $\mathrm{p}\left(x\right)=\left|2x\right|,x\le 0$.

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

Show that ${\mathrm{p}}^{-1}\left(x\right)=-\frac{1}{2}\mathrm{p}(-\frac{1}{2}x).$

The path of a swing boat fairground ride that swings forwards and backwards is modelled as a semi-circle, radius 10 m, as shown in the diagram below.

At time $t$ seconds, the $x$-coordinate of the boat is modelled by the function

$x\left(t\right)=10\sin \left(\frac{\mathrm{\pi }}{5}t\right),t\ge 0,$

and the height, $h$ m, of the boat above the ground, at time $t$seconds, is modelled by

$h\left(t\right)=12-10\left|\cos \left(\frac{\mathrm{\pi }}{5}t\right)\right|,t\ge 0.$

Verify that the initial position of the boat is $(0,2)$.

(i) Write down the coordinates of the boat when it is at its maximum height.

(ii) Find the time it takes the boat to swing between these two points.

Find the position of the boat when it has swung through an angle of  $\frac{\mathrm{\pi }}{6}$ anticlockwise from the $y$-axis, as shown in the diagram above. 

Find the time at which the boat first reaches this position.

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

Solve the equation  $\left|x^2-9\right|=6-0.25x^2$, giving your answers in exact form.

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

$$\begin{gathered} \mathrm{f}\left(x\right)=\left|x^3-8\right|x\in \mathrm{ℝ} \\ \mathrm{g}\left(x\right)=\left|x\right|x\in \mathrm{ℝ} \end{gathered}$$

Sketch the graph of  $y=\mathrm{fg}\left(x\right)$, stating the coordinates of all points where the graph intercepts the coordinate axes.

There are between 0 and 4 solutions to the equation  $\mathrm{fg}\left(x\right)=c$, where $c$ is a real number.  Determine the values of $c$ that produce each number of solutions.

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

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

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

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

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

A sketch of the graph with equation $y=f\left(x\right)$ where $f\left(x\right)=10x-x^2-16$ is shown below.

Points $A$ and $B$ are the $x$-axis intercepts and point $C$ is the maximum point on the graph.

On the diagram above, sketch the graph of $y=-\left|\frac{1}{4}\mathrm{f}\left(\frac{1}{2}x\right)\right|$ labelling the image of the points $A,B$ and $C$ with $A\text{'},B\text{'}\mathrm{and}C\text{'}$.

Show that the area of triangle $ABC$is twice the area of triangle $A\text{'}B\text{'}C\text{'}$.

The function $\mathrm{f}\left(x\right)=e^{3x}-x-6$ is transformed by a sequence of transformations as described below.

1. Horizontal stretch by scale factor 3,

2. The modulus of the function is then taken,

3. Reflection in the $y$-axis.

Write down the resulting transformation in terms of $\mathrm{f}\left(x\right)$ as well as an expression in terms of $x$.

A swing boat fairground ride is modelled as moving forwards and backwards along the path of a semi-circle, radius 18 m, as shown in the diagram below.

Show that, for $0\le \theta \le \frac{\mathrm{\pi }}{2},$

(i) the $x$-coordinate of the boat is given by  $x=18\sin \theta$ ,

(ii) the $y$-coordinate is given by  $y=20-18\cos \theta$

The model is refined so that the coordinates of the boat can be calculated from the time, $t$ seconds, after the boat is set in motion.   The $x$ and $y$ coordinates are now given by 

$x=18\sin Bty=20-18\left|\cos Bt\right|$

where $B$ is a constant.

i) Briefly explain why the modulus of $\cos \theta$ is required for the $y$- coordinate.

ii) Given that the time between the boat reaching its maximum height at either end of the ride is 8 seconds, find the value of $B$.

For $0\le t\le 4$, find the times when the boat is equidistant from the ground and horizontally from the origin.