The function $f$ is defined by

where $x$ is in the largest possible real domain.

A machine produces a component whose mass after $t$ minutes is modelled by

The energy, $E$ joules, required to heat a component of mass $m$ grams is modelled by

The function $f$ is defined by

The function $f$ is defined for $x\in\mathbb{R}$ by

It is known that $f$ is an even function, $f(0)=5$ and $f(\pi)=1$.

The function $f$ is defined by

Let $f(x)=x^5+ax^3+bx^2+cx$, where $a,b,c\in\mathbb{R}$. The function $f$ is odd. Also, $f(1)=0$ and $f(2)=18$.

Let $f(x)=mx+c$, where $m,c\in\mathbb{R}$. The function $f$ is self-inverse, $f$ is not the identity function, and $f(3)=1$.

The function $f$ is defined by

The function $f$ is defined by $f(x)=x^2-4x+1$, for $x\le 2$.

The function $f$ is defined by

Two quantities, $A$ and $B$, are modelled by

where $t$ is measured in hours and $0\leq t\leq 20$.

The function $f$ is defined by

The function $h$ is defined for $x\in\mathbb{R}$ by

It is known that $h$ is an odd function, $h\left(\frac{\pi}{2}\right)=1$ and $h\left(\frac{\pi}{6}\right)=5$.

A linear function $f$ is defined by $f(x)=ax+b$, where $a,b\in\mathbb{R}$. It is known that $f(2)=7$ and $f(6)=-1$.

The function $f$ is defined by $f(x)=3-\sqrt{2x+4}$, for $x\geq -2$.

The function $f$ is defined by
$f(x)=x+2$ for $-3\leq x\leq 1$, and $f(x)=2x+1$ for $1<x\leq 4$.

Let $f(x)=x-1$, where $x\in\mathbb{R}$, and let $g(x)=\sqrt{x+3}$, where $x\geq -3$.

A straight cable is to be installed above a curved roof. The cable passes through the points $A(0,18)$ and $B(60,42)$, where coordinates are measured in metres. The height of the roof above the same horizontal axis is modelled by

The support strut is a straight line that passes through the point $C(45,62)$ and is perpendicular to the cable.

The function $f$ is defined by

where $x$ is in the largest possible real domain.

A heating process is modelled by the function

where $T$ is the temperature in degrees Celsius after $t$ minutes. The quality score of the product at temperature $T$ is modelled by

The function $f$ is defined for $x\in\mathbb{R}$ by

The function is one-to-one.

For a real constant $a$, the function $f$ is defined by

For $a\in\mathbb{R}$, define the function $f_a:\mathbb{R}\to\mathbb{R}$ by $f_a(x)=a-x$. Also define $f_0(x)=-x$ and, for $n\in\mathbb{Z}^+$, let $T_a^n$ denote applying $T_a$ repeatedly $n$ times, where $T_a=f_a\circ f_0$.

A function $f$ is defined on a domain symmetric about $0$. Define

The functions $E$ and $O$ are called the even and odd parts of $f$.

Two linear functions are defined by $f(x)=mx+c$ and $g(x)=px+q$, where $m,p,c,q\in\mathbb{R}$. This question investigates when the order of two linear calibrations matters.

Let $a>0$. Define

and

The two functions appear to undo each other, but only in one order without further restriction.

Let $a>0$ and define

The largest possible real domain is used unless otherwise stated.

A function on a finite set can be represented by a mapping table. Let $S=\{1,2,3,4\}$. Functions $f:S\to S$ and $g:S\to S$ are given by the table.

Let $f(x)=x^2-2x-3$ and $g(x)=x+1$, for $x\in\mathbb{R}$. The function $d$ is defined by $d(x)=f(x)-g(x)$.