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

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 $$x \in \mathbb{R}, x \ne 0$$.


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


 * 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, $$f(x)=x^2, x \in \mathbb{R}$$ is not a one-to-one function, but $$f(x)=x^2, x \geq 0$$ 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


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 $$f(x) = \frac{1}{x}, x \geq 1$$

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 $$0 < f(x) \leq 1$$. This range can also be expressed in interval notation as $$f(x) \in (0,1]$$



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

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.

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

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 $$g(x) \geq -\frac{9}{4}$$, which can be expressed as $$g(x) \in [-\frac{9}{4},\infty)$$

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 $$g(x) = 2x^2$$.

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

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 $$f(x)=f(y) \implies x = y$$

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

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

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 $$f(x) = (2x+3)^2 -4$$

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

Graphing Inverse Functions


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.