0 - Functions

A function is a mapping between sets. The notation is as follows:

Are notationally different but mean the same.

• domain (inputs), $x \in \mathbb{R}$
• range (outputs), $f(x) \in \mathbb{R}$

All functions are mappings, but not all mappings are functions.

• one-to-one mapping: one value of x mapped to one value of f(x)
• one-to-many mapping: one value of x mapped to many values of f(x)
• many-to-one mapping: many values of x mapped to one value of f(x)

All functions are mappings, but not all mappings are functions.

The definition of function states that for each member of the domain there can be only one member of the range. Thus the graph of a function cannot look like this:

0.1 - Finding the Range

$f(x)$ has domain $x \in \mathbb{R} \\ f(x) = x^2 + 3x - 1$

Can be found by either completing the square or graphically.

0.1.2 - Completing the square

$(x + \frac{3}{2})^2 - (\frac{3}{2})^2 - 1 \\ \Rightarrow 0 - \frac{13}{4} \\ \\ \therefore f(x) \geq \frac{-13}{4} \rightarrow range$

0.1.2- Graphically

Visually see the minimum value of y does not go below $\frac{-13}{4}$, one can consider the x axis as our domain and y axis the range.

For a restricted domain one can only consider the values of f(x) between the mentioned domain, e.g. f(x) has domain $-3 \geq x \geq 5$

1 - Inverse Functions

Notation is as follows:

When asked to form an inverse function from a function, one can swap variables for some arbitrary value and solve for x, e.g.

The inverse function also has an inverted domain, range and mapping.

A function can only be a function if it has a one-to-one mapping. The inverse of a function is a reflection of the function in the line $y=x$.

2 - Composite Functions

Essentially $gh(x)=g(h(x))$ and $hg(x)=h(g(x))$, order matters.

3 - Modulus Equations

Basically math.abs().

\          |   /
\         |  /
\        | /                             |  y=sin(|x|)
\       |/                              |
\      +-1                      .--.   |   .--.
\    /|                       /    \  |  /    \
\  / | y=|2x+1|             /      \ | /      \
_______\/__+_____     ___________;________;+;________;___________
-.5  |            \       /          |          \       /
|             \     /           |           \     /
|              '._.'            |            '._.'


3.1 - Solving mod equations

3.1.1 - Linear

Square both sides.

3.1.2 - Non-linear

Graphically.

60 +--------------------------------------------------+
|*           +            +           +            |
|*                        abs((x*x)-20)-20 ********|
|*                                  3*x-10 #######*|
|     *                                       *    |
20 |-+   *                                      *   +-|
|     *                                      *   ##|
|     *                                      *  ## |
|      *                                     *##   |
|      *                                    *##    |
|      *                                   ##      |
|       *                                 ##*      |
|       *                                ##*       |
|       *                              ##  *       |
|        *                            ##   *       |
|        *                          ##    *        |
|        *                         ##     *        |
0 |-+       *            ******    ##       *      +-|
|         *          **      **##        *         |
|         *         **        ##         *         |
|          *       **       ## **       *          |
|          *      **       ##   **      *          |
|          *      *      ##      *      *          |
|           *    *      ##        *    *           |
|           *   *     ##           *   *           |
|            * *     ##             * *            |
|            * *   ##               * *            |
|             *   ##                 *             |
-20|-+             ##                               +-|
|             ##                                   |
|            +            +           +            |
-40+--------------------------------------------------+
-10          -5            0           5            10


The line $y=6x$ intersects the curve at two points, these are our solutions.

In the region in which the line is inverted because of the modulus, $-5 \lt x \lt 5$, the function must also be inverted, $y=-x^2+1$, we can then use this to solve for the intersection between the two lines. In the region in which the quadratic graph is not inverted, we can just solve for the intersection normally.

Using this, the two solutions for $y=\|x^2-1\|$ and $y=6x$ are $\pm3\sqrt{10}$.

4 - Natural log and $e^x$

$ln$ and $e$ are the inverse of each other.

$ln$ is a logarithm and therefore is still subjected to the laws of logs:

$e$, Euler’s number is irrational, $e\approx2.718281$

5 - Transformations

• $-$: reflection about $x=0$ (in the y axis)
• $a$: stretch in $y$ scale factor $a$
• $-$: reflection about $y=0$ (in the x axis)
• $b$: stretch in $x$ scale factor $\frac{1}{b}$
• $+c$: translation in $x -c$
• $+d$: translation in $y \text{ }d$

6 - cot, sec and cosec

Some identities can be derived from C2’s $sin^2x+cos^2x=1$:

7 - Differentiating $e^x$

Simply multiply by the differential of the index.

8 - Integrating $e^x$

Divide by the differential of the index.

9 - Differentiating & integrating sin and cos

Remember that integration and differentiation are basically the inverse of each other.

10 - Chain Rule

Used for differentiating a function of a function, e.g. of the form $f(g(x))$.

$du$’s cancel out: $\require{cancel} \frac{dy}{\cancel{du}} \times \frac{\cancel{du}}{dx}$

10.1 - Chain Rule example

Functions in a function in a function will require the chain rule to be performed twice, and so forth for further nested functions.

11 - Product Rule

11.1 - Product Rule example

This example showcases use of the Chain Rule also.

15 - Intergration of the form $\frac{f'(x)}{f(x)}$

Transform the integral into the form:

Where $a$ is typically a constant to make $\frac{f'(x)}{f(x)}$ true.

17 - Integration by Substitution

Introduces a variable in order to simplify the integration (similar to the Chain Rule).

18 - Simpsons Rule

• $\sum odd$: Sum of all odd ordinates
• $\sum even$: Sum of all even ordinates
• $h$: Strip width
• $n$: Number of strips

in x

in y