A-level Mathematics/CIE/Pure Mathematics 1/Functions

Notation & Terms

Function Notation

There are two main ways of describing a function: with parenthesis notaton, e.g. $f(x)=x^2-1$, or with mapping notation, e.g. $f:x\mapsto x^2-1$. These both describe what the function does.

Definitions

Function

A function is a mapping from an input set to an output set. For example, the function $f(x)=3x$ maps the input $x$ to the output $3x$ by multiplying it by 3

Domain

The domain is the set of all valid inputs. e.g., the function $f(x)=\frac{1}{x}$ has the domain $x\in ℝ,x\ne 0$.

Range

The range is the set of all possible outputs. e.g., the function $f(x)=x^2$ has the range $x\ge 0$

One-to-one function

A one-to-one function is a function which maps each input to exactly one output, and each output corresponds to exactly one input. For instance, $f(x)=x^2,x\in ℝ$ is not a one-to-one function, but $f(x)=x^2,x\ge 0$ is a one-to-one function.

Inverse function

An inverse function is a function that does the opposite of another function. For example, the function $f(x)=x+5$ has an inverse function $f^{-1}(x)=x-5$.

Composition of functions

Composition of functions is where the output of one function is input into another function. For example: if $f(x)=x^2$ and $g(x)=3x+5$, the composed function $fg(x)=(3x+5{)}^2$ and the composed function $gf(x)=3x^2+5$. Note that for two arbitrary functions $f(x)$ and $g(x)$, the composed functions $fg(x)$ and $gf(x)$ are not equal except for some special cases.

Finding the Range

The function y=1/x approaches zero as x goes to infinity.

To find the range of a function, we need to find the highest and lowest values that the function can take.

Example 1

Find the range of $f(x)=\frac{1}{x},x\ge 1$

The lowest value that $x$ can take is 1, so one bound of the range is $f(x)=\frac{1}{1}=1$.

The highest value that $x$ can take is infinite, so the other bound of the range is the value that $f(x)$ approaches as $x$ goes to infinity, which is 0.

So the range of the function $f(x)$ is $0<f(x)\le 1$. This range can also be expressed in interval notation as $f(x)\in (0,1]$

The vertex is where the graph reaches its lowest point.

Example 2

Find the range of $g(x)=x^2-x-2,x\in ℝ$

A quadratic function always has a turning point, known as its vertex. This determines its range. The vertex can be found by completing the square.

$\begin{array}{cc}g(x)& =x^2-x-2\\ & =(x-\frac{1}{2}{)}^2-\frac{1}{4}-2\\ & =(x-\frac{1}{2}{)}^2-\frac{9}{4}\end{array}$

Completing the square provides the coordinates of the vertex, $(\frac{1}{2},-\frac{9}{4})$.

Since the vertex is the lowest point of this function, we can express the range as $g(x)\ge -\frac{9}{4}$, which can be expressed as $g(x)\in [-\frac{9}{4},\mathrm{\infty })$

Composing Functions

A composite function is a function which is created by taking the output of one function as the input of another function.

e.g. Find the composite function $gf(x)$ when $f(x)=3x+5$ and $g(x)=2x^2$.

$\begin{array}{cc}gf(x)& =g(f(x))=g(3x+5)\\ & =2(3x+5{)}^2=2(9x^2+30x+25)\\ & =18x^2+60x+50\end{array}$

It is important to note that a composite function can only be created if the range of the inner function is within the domain of the outer function.

Inverse Functions

An inverse function is the reverse of a given function, such as how $f(x)=x+5$ has the inverse $f^{-1}(x)=x-5$.

However, not all functions have an inverse. Only one-to-one functions have an inverse.

Determining whether a function is one-to-one

The formal way of determining whether a function is one-to-one is to prove that $f(x)=f(y)⟹x=y$

e.g. Prove that $f(x)=x^3$ is one-to-one.

$\begin{array}{cc}f(x)& =f(y)\\ x^3& =y^3\\ \sqrt[3]{x^3}& =\sqrt[3]{y^3}\\ x& =y\\ \therefore f(x)\text{ is one-to-one}\end{array}$

Finding the inverse

To find the inverse of a function, substitute $x$ for $f^{-1}(x)$ in the function definition then rearrange the variables to make $f^{-1}(x)$ the subject of the formula.

e.g. Find the inverse of $f(x)=(2x+3{)}^2-4$

$\begin{array}{c}f(f^{-1}(x))=(2f^{-1}(x)+3{)}^2-4\\ x=(2f^{-1}(x)+3{)}^2-4\\ (2f^{-1}(x)+3{)}^2=x-4\\ 2f^{-1}(x)+3=\sqrt{x-4}\\ 2f^{-1}(x)=\sqrt{x-4}-3\\ f^{-1}(x)=\frac{\sqrt{x-4}-3}{2}\end{array}$

Graphing Inverse Functions

A graph of a function and its inverse

If you plot a function and its inverse on the same graph, it is apparent that the graph of the inverse is the same as the graph of the function reflected across the line $y=x$.

The reason for this is that the graph $y=f^{-1}(x)$ is equivalent to $x=f(y)$

Transforming Functions

A transformation of a function changes the position, size, or shape of the function's graph.

Translation

Translation changes the position of the function's graph. $y=f(x)+a$ can move the function vertically and $y=f(x+a)$ can move the function horizontally.

Scaling

Scaling is where the function changes in size. $y=af(x)$ changes the size vertically and $y=f(ax)$ changes the size horizontally. If $a$ is negative, the function will also be reflected.

Reflection

Reflection is where the function is mirrored across a given line. This can be achieved with $y=a-f(x)$ for vertical reflection and $y=f(a-x)$ for horizontal reflection.

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