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

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

$\begin{array}{lcccc}\mathrm{f}\left(x\right)=7-2x^2& & x\in \mathrm{ℝ}& & \\ \mathrm{g}\left(x\right)=\frac{3x}{5x-1}& & x\in \mathrm{ℝ}& & x\ne \frac{1}{5}\end{array}$

State the range of $\mathrm{f}$

Find $\mathrm{gf}\left(1.8\right)$

Find ${\mathrm{g}}^{-1}\left(x\right)$

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of the graph with equation $y=\left|3-2x\right|$

Solve

$\left|3-2x\right|=7+x$

The point $P(-2,-5)$ lies on the curve with equation $y=\mathrm{f}\left(x\right),x\in \mathrm{ℝ}$

Find the point to which $P$ is mapped, when the curve with equation $y=\mathrm{f}\left(x\right)$ is transformed to the curve with equation

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

(i) Sketch the graph of $y=\left|x\right|+3$.

(ii) Explain why $\left|x\right|+3\ge \left|x+3\right|$ for all real values of $x$.

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

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

Sketch the 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 $\mathrm{f}$.

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 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 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

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

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.

$\mathrm{g}\left(x\right)=\frac{2x+5}{x-3}x\ge 5$

Find $\mathrm{gg}\left(5\right)$.

State the range of $\mathrm{g}$.

Find ${\mathrm{g}}^{-1}\left(x\right)$, stating its domain.

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

$\mathrm{f}\left(x\right)=\frac{8x+5}{2x+3}x>-\frac{3}{2}$

Find ${\mathrm{f}}^{-1}\left(\frac{3}{2}\right)$

Show that

$\mathrm{f}\left(x\right)=A+\frac{B}{2x+3}$

where $A$ and $B$ are constants to be found.

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

$\mathrm{g}\left(x\right)=16-x^{2}0\le x\le 4$

State the range of ${\mathrm{g}}^{-1}$

Find the range of $\mathrm{f}{\mathrm{g}}^{-1}$

Figure 4 shows a sketch of the graph of $y=\mathrm{g}\left(x\right)$, where

$\mathrm{g}\left(x\right)=\{\begin{array}{ll}{\left(x-2\right)}^2+1& x\le 2\\ 4x-7& x>2\end{array}$

Find the value of $\mathrm{gg}\left(0\right)$.

Find all the values of $x$ for which

$\mathrm{g}\left(x\right)>28$

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

$\mathrm{h}\left(x\right)={\left(x-2\right)}^2+1x\le 2$

Explain why $\mathrm{h}$ has an inverse but $\mathrm{g}$ does not.

Solve

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

Figure 2 shows a sketch of the graph with equation

$y=2\left|x+4\right|-5$

The vertex of the graph is at the point $P$, shown in Figure 2.

Find the coordinates of $P$.

Solve the equation

$3x+40=2\left|x+4\right|-5$

A line $l$ has equation $y=ax$, where $a$ is a constant.

Given that $l$ intersects $y=2\left|x+4\right|-5$ at least once,

find the range of possible values of $a$, writing your answer in set notation.

$\begin{array}{ccc}\mathrm{f}\left(x\right)=2x^2+4x+9& & x\in \mathrm{ℝ}\end{array}$

Write $\mathrm{f}\left(x\right)$ in the form $a{\left(x+b\right)}^2+c$, where $a,b$ and $c$ are integers to be found.

Sketch the curve with equation $y=\mathrm{f}\left(x\right)$ showing any points of intersection with the coordinate axes and the coordinates of any turning point.

Find the range of the function

$\begin{array}{ccc}\mathrm{h}\left(x\right)=\frac{21}{2x^2+4x+9}& & x\in \mathrm{ℝ}\end{array}$

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

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