Functions

The function $f$ is defined by $f(x) = a^2x^2 - ax + 3b$ for $x \leq \frac{1}{2a}$, with $a$ and $b$ as constants.

Feb/March 2016

For $x \geq 0$, the functions $f$ and $g$ are given by $f : x \mapsto 2x^2 + 3$ and $g : x \mapsto 3x + 2$.

Feb/March 2017

Express the quadratic $x^2 - 4x + 7$ in the form $(x + a)^2 + b$.

Feb/March 2019

The graph of $y = f(x)$ is mapped onto the graph of $y = 1 + f\left(\tfrac{1}{2}x\right)$.

Feb/March 2020

Write $2x^2 + 12x + 11$ in the form $2(x + a)^2 + b$, where $a$ and $b$ are constants.

Feb/March 2020

The diagram presents the graph of $y = f(x)$ using solid lines. The graph drawn with broken lines is a transformed version of $y = f(x)$.

Feb/March 2021

Functions $f$ and $g$ are given by: $f: x \mapsto x^2 + 2x + 3$ when $x \leq -1$, $g: x \mapsto 2x + 1$ when $x \geq -1$.

Feb/March 2021

Rewrite $2x^2 - 8x + 14$ in the form $2[(x - a)^2 + b]$.

Feb/March 2022

The function $f$ is specified by $f(x) = -3x^2 + 2$ for $x \leq -1$.

Feb/March 2023

The diagram shows a section of the curve with equation $y = k\sin\left(\frac{1}{2}x\right)$, where $k$ is a positive constant and $x$ is measured in radians. Point $A$ is a minimum point on the curve.

Feb/March 2024

The functions $f$ and $g$ are given for every real value of $x$ by $f(x) = (3x - 2)^2 + k$ and $g(x) = 5x - 1$, with $k$ a constant.

Feb/March 2024

For every real value of $x$, the functions f and g are defined by $f(x) = 4x^2 - c$ and $g(x) = 2x + k$, where $c$ and $k$ are positive constants. It is stated that $g^{-1}(3k + 1) = c$.

Feb/March 2025

Consider the function $f: x \mapsto 4 - 3\sin x$, defined over the domain $0 \leq x \leq 2\pi$.

May/June 2010

For $x \in \mathbb{R}$, the functions $f$ and $g$ are given by $f: x \mapsto 4x - 2x^2$, $g: x \mapsto 5x + 3$.

May/June 2010

Functions $f$ and $g$ are defined on $x \in \mathbb{R}$ by $f : x \mapsto 2x + 1$, $g : x \mapsto x^2 - 2$.

May/June 2011

The function $f$ is given by $f : x \mapsto \frac{x + 3}{2x - 1}$, $x \in \mathbb{R}$, $x \neq \tfrac{1}{2}$.

May/June 2011

The function $f$ is specified by $f(x) = 3 - 4\cos^k x$, for $0 \leq x \leq \pi$, and $k$ is a constant.

May/June 2011

The functions $f$ and $g$ are specified by $f : x \mapsto 3x - 4$, $x \in \mathbb{R}$, and $g : x \mapsto 2(x - 1)^3 + 8$, $x > 1$.

May/June 2011

The function $f : x \mapsto x^2 - 4x + k$ is given on the domain $x \geq p$, with $k$ and $p$ as constants.

May/June 2012

The functions $f$ and $g$ are specified by $f : x \mapsto 2x + 5$ for $x \in \mathbb{R}$, and $g : x \mapsto \frac{8}{x - 3}$ for $x \in \mathbb{R}, x \neq 3$.

May/June 2012

Rewrite $2x^2 - 12x + 13$ in the form $a(x + b)^2 + c$, where $a$, $b$ and $c$ are constants.

May/June 2013

The function $f$ is given by $f : x \mapsto 2x + k$, $x \in \mathbb{R}$, where $k$ is a fixed constant.

May/June 2013

The diagram represents the function $f$ on $-1 \leq x \leq 4$, where $$f(x) = \begin{cases}3x - 2 & \text{for } -1 \leq x \leq 1, \\ \frac{4}{5 - x} & \text{for } 1 < x \leq 4.\end{cases}$$

May/June 2014

Functions $f$ and $g$ are given by $f : x \mapsto 2x - 3$, $x \in \mathbb{R}$, and $g : x \mapsto x^2 + 4x$, $x \in \mathbb{R}$. The function $h$ is given by $h : x \mapsto x^2 + 4x$ for $x \geq k$, and it is stated that $h$ is invertible.

May/June 2014

The function $f$ is given by $f(x)=\frac{15}{2x+3}$ for $0 \le x \le 6$.

May/June 2014

The function $f : x \mapsto 5 + 3\cos\left(\frac{1}{2}x\right)$ is defined over the interval $0 \le x \le 2\pi$.

May/June 2015

The function $f$ satisfies $f'(x) = 5 - 2x^2$, and the point $(3, 5)$ lies on the graph of $y = f(x)$.

May/June 2015

The mapping $f$ is defined, for $x \in \mathbb{R}$, by $f: x \mapsto 2x^2 - 6x + 5$.

May/June 2015

The diagram presents the graph of $y = f^{-1}(x)$, where $f^{-1}$ is given by $f^{-1}(x) = \frac{1 - 5x}{2x}$ for $0 < x \leq 2$.

May/June 2015

For $-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi$, the function $f$ is specified by $f : x \mapsto 4\sin x - 1$.

May/June 2016

The functions $f$ and $g$ are given by $f : x \mapsto 10 - 3x,\ x \in \mathbb{R}$, and $g : x \mapsto \frac{10}{3 - 2x},\ x \in \mathbb{R},\ x \ne \frac{3}{2}$.

May/June 2016

The function $f$ is given by $f : x \mapsto 6x - x^2 - 5$ for all $x \in \mathbb{R}$.

May/June 2016

The function $f$ is defined by $f(x) = 2x + 3$ when $x \geq 0$. The function $g$ is defined by $g(x) = ax^2 + b$ for $x \leq q$, where $a$, $b$ and $q$ are constants. The composite function $fg$ satisfies $fg(x) = 6x^2 - 21$ for $x \leq q$.

May/June 2016

The mapping $f$ is defined as $f : x \mapsto \dfrac{2}{3 - 2x}$ for $x \in \mathbb{R}$, $x \neq \dfrac{3}{2}$.

May/June 2017

The function $f$ has rule $f(x) = 3 \tan\left(\frac{1}{2}x\right) - 2$, for $-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi$.

May/June 2017

Find the coordinates of the points where the curve $y = x^{\frac{2}{3}} - 1$ intersects the curve $y = x^{\frac{1}{3}} + 1$.

May/June 2017

Express $9x^2 - 6x + 6$ in the form $(ax + b)^2 + c$, where $a$, $b$ and $c$ are constants.

May/June 2017

For $x \in \mathbb{R}$, the functions $f$ and $g$ are given by $f : x \mapsto \frac{1}{2}x - 2$, $g : x \mapsto 4 + x - \frac{1}{2}x^2$.

May/June 2018

The function $f$ is defined by $f(x) = a + b\cos x$ for $0 \leq x \leq 2\pi$. It is given that $f\left(\tfrac{1}{3}\pi\right) = 5$ and $f(\pi) = 11$.

May/June 2018

The function $f$ is one-one and is given by $f(x) = (x - 2)^2 + 2$ for $x \geq c$, where $c$ is a constant.

May/June 2018

For $0 \leq x \leq 2\pi$, the function $f$ is given by $f(x) = 2 - 3 \cos x$.

May/June 2019

The functions $f$ and $g$ are given by $f: x \mapsto 3x - 2, \; x \in \mathbb{R}$ and $g: x \mapsto \frac{2x + 3}{x - 1}, \; x \in \mathbb{R}, \; x \ne 1$.

May/June 2019

The function $f$ is given by $f(x) = \frac{48}{x - 1}$ for $3 \leq x \leq 7$. The function $g$ is given by $g(x) = 2x - 4$ for $a \leq x \leq b$, where $a$ and $b$ are constants.

May/June 2019

The function $f : x \mapsto p\sin^2 2x + q$ is defined on $0 \leq x \leq \pi$, with $p$ and $q$ being positive constants. The diagram displays the graph of $y = f(x)$.

May/June 2019

The diagram presents the graph of $y = f(x)$, where $f(x) = \frac{3}{2}\cos 2x + \frac{1}{2}$ for $0 \leq x \leq \pi$.

May/June 2020

The functions $f$ and $g$ are given for $x \in \mathbb{R}$ by $f: x \mapsto \frac{1}{2}x - a$ and $g: x \mapsto 3x + b$, with $a$ and $b$ constant.

May/June 2020

For $x \in \mathbb{R}$, the function $f$ is defined by $f: x \mapsto a - 2x$, where $a$ is a constant.

May/June 2020

Let functions $f$ and $g$ be defined by $f(x) = 2 - 3\sin 2x$ for $0 \leq x \leq \pi$, and by $g(x) = -2f(x)$ for $0 \leq x \leq \pi$.

May/June 2020

In each of parts (a), (b) and (c), the graph drawn with solid lines has equation $y = f(x)$. The graph drawn with broken lines is a transformed version of $y = f(x)$.

May/June 2020

The functions $f$ and $g$ are given by $f(x) = x^2 - 4x + 3$ for $x > c$, where $c$ is a constant, and $g(x) = \frac{1}{x+1}$ for $x > -1$.

May/June 2020

The functions $f$ and $g$ are given by: $f(x) = (x - 2)^2 - 4$ for $x \geq 2$, $g(x) = ax + 2$ for $x \in \mathbb{R}$, with $a$ as a constant.

May/June 2021

The graph of $y = f(x)$ becomes $y = 2f(x - 1)$. Describe fully the two separate transformations that have been combined to produce the final image.

May/June 2021

The function $f$ is given by $f(x) = 2x^2 + 3$ for $x \geq 0$.

May/June 2021

Functions $f$ and $g$ are each defined for $x \in \mathbb{R}$, and they are given by $f(x) = x^2 - 2x + 5$, $g(x) = x^2 + 4x + 13$.

May/June 2021

The functions $f$ and $g$ are given by $f : x \mapsto x^2 - 1$ when $x < 0$, and $g : x \mapsto \frac{1}{2x + 1}$ when $x < -\frac{1}{2}$.

May/June 2021

The function $f$ is given by $f(x) = \dfrac{x^2 - 4}{x^2 + 4}$, with the condition $x > 2$.

May/June 2022

The functions $f$ and $g$ are given by $f(x) = \frac{2x + 1}{2x - 1}$ for $x \neq \frac{1}{2}$, $g(x) = x^2 + 4$ for $x \in \mathbb{R}$. The diagram shows a section of the graph of $y = f(x)$.

May/June 2022

The curve given by $y = x^2 + 2x - 5$ is translated by $\begin{pmatrix}-1\\3\end{pmatrix}$. Find the equation of the translated curve, and give your answer in the form $y = ax^2 + bx + c$.

May/June 2022

The function $f$ is given by $f(x) = 2x^2 - 16x + 23$ for $x < 3$.

May/June 2022

The graphs of $y = f(x)$ and $y = g(x)$ are shown in the diagram.

May/June 2023

The functions $f$ and $g$ are specified below, with $a$ and $b$ taken as constants. $f(x) = 1 + \frac{2a}{x - a}$ for $x > a$ and $g(x) = bx - 2$ for $x \in \mathbb{R}$.

May/June 2023

The diagram presents the graph of $y = f(x)$, where $f$ is defined by $f(x) = 3 + 2\sin\left(\frac{1}{4}x\right)$ for $0 \leq x \leq 2\pi$.

May/June 2023

The diagram displays the graph of $y = f(x)$, made up of the two straight lines $AB$ and $BC$. The lines $A'B'$ and $B'C'$ give the graph of $y = g(x)$, obtained by carrying out a sequence of two transformations, in either order, on $y = f(x)$.

May/June 2023

The function $f$ is given by $f(x) = 2 - \frac{5}{x + 2}$, with $x > -2$.

May/June 2023

The diagram contains two curves. One has equation $y = \sin x$, while the other has equation $y = f(x)$.

May/June 2024

The function $f$ is given by $f(x)=\dfrac{2}{x^2}+4$ for $x<0$, and the diagram displays the graph of $y=f(x)$.

May/June 2024

The curve $y = x^2$ is mapped onto the curve $y = 4(x - 3)^2 - 8$.

May/June 2024

The function $f$ is specified by $f(x) = \sqrt{x} - 1$ for $x > 1$.

May/June 2024

The function $f$ is given by $f(x) = 10 + 6x - x^2$ for all $x \in \mathbb{R}$.

May/June 2024

A curve goes through the point $\left(\frac{4}{5}, -3\right)$ and has $\frac{dy}{dx} = \frac{-20}{(5x - 3)^2}$.

May/June 2024

The functions $f$ and $g$ are given by $f(x)=\sqrt{x}$ for $x\geq 0$, and $g(x)=3\sqrt{x+2}-5$ for $x\geq -2$.

May/June 2025

The diagram displays the graphs of the equations $y = f(x)$ and $y = g(x)$.

May/June 2025

Write $x^2 + 4x + 2$ in the form $(x + a)^2 + b$, where $a$ and $b$ are integers.

May/June 2025

The function $f$ is given by $f(x) = x^2 + 4ax + a$ for every $x \in \mathbb{R}$, with $a$ a constant. The function $g$ is defined so that $g^{-1}(x) = \sqrt[3]{2x - 4}$ for every $x \in \mathbb{R}$.

May/June 2025

The functions $f$ and $g$ are given by $f(x) = \cos x$ for $0 \leq x \leq \pi$, and by $g(x) = 3\cos(x - \pi) + 2$ for $\pi \leq x \leq 2\pi$.

May/June 2025

The functions $f$ and $g$ are given for $x \in \mathbb{R}$ by $f : x \mapsto 2x + 3$ and $g : x \mapsto x^2 - 2x$.

Oct/Nov 2010

The function $f$ is specified by $f : x \mapsto 3 - 2\tan\left(\frac{1}{2}x\right)$ for $0 \leq x < \pi$.

Oct/Nov 2010

The function $f$ is given by $f(x) = x^2 - 4x + 7$ when $x > 2$.

Oct/Nov 2010

The diagram represents the function $f$ on $0 \leq x \leq 6$, with $x \mapsto \frac{1}{2}x^2$ for $0 \leq x \leq 2$ and $x \mapsto \frac{1}{2}x + 1$ for $2 < x \leq 6$.

Oct/Nov 2010

The functions $f$ and $g$ are given by $f: x \mapsto 2x^2 - 8x + 10$ for $0 \leq x \leq 2$, and $g: x \mapsto x$ for $0 \leq x \leq 10$.

Oct/Nov 2011

The functions $f$ and $g$ are defined for $x \in \mathbb{R}$ by $f: x \mapsto 3x + a$ and $g: x \mapsto b - 2x$, where $a$ and $b$ are constants. It is given that $f(2) = 10$ and $g^{-1}(2) = 3$.

Oct/Nov 2011

The functions $f$ and $g$ are given by $f : x \mapsto 2x + 3$ for $x \leq 0$, and $g : x \mapsto x^2 - 6x$ for $x \leq 3$.

Oct/Nov 2011

The function $f$ is defined by $f(x) = 4x^2 - 24x + 11$, where $x \in \mathbb{R}$.

Oct/Nov 2012

For $x \geq -3$, the function $f$ satisfies $f(x) = \sqrt{\left(\frac{x+3}{2}\right)} + 1$.

Oct/Nov 2012

For $-\tfrac{1}{2}\pi \leq x \leq \tfrac{1}{2}\pi$, the functions $f$ and $g$ are given by $f(x) = \tfrac{1}{2}x + \tfrac{1}{6}\pi$ and $g(x) = \cos x$.

Oct/Nov 2012

The diagram displays a section of the curve $y = 11 - x^2$ together with a segment of the straight line $y = 5 - x$, and they intersect at the point $A(p, q)$, where $p$ and $q$ are positive constants.

Oct/Nov 2012

A curve is described by the equation $y = f(x)$. It is stated that $f'(x) = \frac{1}{\sqrt{x + 6}} + \frac{6}{x^2}$ and that $f(3) = 1$.

Oct/Nov 2013

The function $f$ is given by $f: x \mapsto x^2 + 1$ for $x \geq 0$.

Oct/Nov 2013

The function $f$ is given by $f : x \mapsto x^2 + 4x$ for $x \geq c$, where $c$ is a constant. It is stated that $f$ is a one-one function.

Oct/Nov 2013

The function $f : x \mapsto 6 - 4\cos\left(\frac{1}{2}x\right)$ is defined over the interval $0 \leq x \leq 2\pi$.

Oct/Nov 2014

For $x \geq 0$, the functions are given by $f: x \mapsto (ax + b)^{\frac{1}{3}}$, where $a$ and $b$ are positive constants, and $g: x \mapsto x^2$. If $fg(1) = 2$ and $fg(9) = 16$, calculate the values of $a$ and $b$.

Oct/Nov 2014

The function $f$ satisfies $f'(x) = 3x^2 - 7$ and $f(3) = 5$.

Oct/Nov 2015

The function $f: x \mapsto -x^2 + 6x - 5$ has domain $x \geq m$, with $m$ as a constant.

Oct/Nov 2015

The functions $f$ and $g$ are given by $f: x \mapsto 3x + 2,\ x \in \mathbb{R}$ and $g: x \mapsto 4x - 12,\ x \in \mathbb{R}$.

Oct/Nov 2015

The rule for $f$ is $f(x) = 3x + 1$ when $x \leq a$, where $a$ is constant. The rule for $g$ is $g(x) = -1 - x^2$ when $x \leq -1$.

Oct/Nov 2015

The functions $f$ and $g$ are specified by $f(x) = \frac{4}{x} - 2$ for $x > 0$, and $g(x) = \frac{4}{5x + 2}$ for $x \geq 0$.

Oct/Nov 2016

Write $4x^2 + 12x + 10$ in the form $(ax + b)^2 + c$, with $a$, $b$ and $c$ as constants.

Oct/Nov 2016

For $x > 3$, the functions $f$ and $g$ are given by $f:x \mapsto \frac{1}{x^2 - 9}$ and $g:x \mapsto 2x - 3$.

Oct/Nov 2017

The function $f$ is given by $f : x \mapsto 4 - 5x$ for $x \in \mathbb{R}$.

Oct/Nov 2017

The function $f$, given by $f : x \mapsto a + b\sin x$ for $x \in \mathbb{R}$, satisfies $f\left(\frac{\pi}{6}\right) = 4$ and $f\left(\frac{\pi}{2}\right) = 3$.

Oct/Nov 2017