A **function** is a rule where each input has exactly one output, which can be determined by the **vertical line test**.
!!! definition
- The **domain** is the set of allowable independent values.
- The **range** is the set of allowable dependent values.
Functions can be **composed** to apply the result of one function to another.
$$
(f\circ g)(x) = f(g(x))
$$
!!! warning
Composition is not commutative: $f\circ g \neq g\circ f$.
## Inverse functions
The inverse of a function swaps the domain and range of the original function: $f^{-1}(x)$ is the inverse of $f(x)$.. It can be determined by solving for the other variable:
$$
\begin{align*}
y&=mx+b \\
y-b&=mx \\
x&=\frac{y-b}{m}
\end{align*}
$$
Because the domain and range are simply swapped, the inverse function is just the original function reflected across the line $y=x$.
<imgsrc="https://upload.wikimedia.org/wikipedia/commons/1/11/Inverse_Function_Graph.png"width=300>(Source: Wikimedia Commons, public domain)</img>
If the inverse of a function is applied to the original function, the original value is returned.
$$f^{-1}(f(x)) = x$$
A function is **invertible** only if it is "**one-to-one**": each output must have exactly one input. This can be tested via a horizontal line test of the original function.
If a function is not invertible, restricting the domain may allow a **partial inverse** to be defined.
!!! example
<imgsrc="https://upload.wikimedia.org/wikipedia/commons/7/70/Inverse_square_graph.svg">(Source: Wikimedia Commons, public domain)</img>
By restricting the domain to $[0,\inf]$, the **multivalued inverse function** $y=\pm\sqrt{x}$ is reduced to just the partial inverse $y=\sqrt{x}$.
## Symmetry
An **even function** satisfies the property that $f(x)=f(-x)$, indicating that it is unchanged by a reflection across the y-axis.
An **odd function** satisfies the property that $-f(x)=f(-x)$, indicating that it is unchanged by a 180° rotation about the origin.
The following properties are always true for even and odd functions:
- even × even = even
- odd × odd = even
- even × odd = odd
Functions that are symmetric (that is, both $f(x)$ and $f(-x)$ exist) can be split into an even and odd component. Where $g(x)$ is the even component and $h(x)$ is the odd component:
$$
\begin{align*}
f(x) &= g(x) + h(x) \\
g(x) &= \frac{1}{2}(f(x) + f(-x)) \\
h(x) &= \frac{1}{2}(f(x) - f(-x))
\end{align*}
$$
!!! note
The hyperbolic sine and cosine are the even and odd components of $f(x)=e^x$.
The Heaviside function effectively returns a boolean whether the number is greater than 0.
$$
\begin{align*}
H(x) = \begin{cases}
0 &\text{ if } t <0 \\
1 &\text{ if } t \geq 0
\end{cases}
\end{align*}
$$
This can be used to construct other piecewise functions by enabling them with $H(x-a)$ as a factor, where $a$ is the interval.
In a nutshell:
- $1-H(t-a)$ lets you "turn a function off" at at $t=a$
- $H(t-a)$ lets you "turn a function on at $t=a$
- $H(t-a) - H(t-b)$ leaves a function on in the interval $(a, b)$
!!! example
TODO: example for converting piecewise to heaviside via collecting heavisides
and vice versa
## Periodicity
The function $f(t)$ is periodic only if there is a repeating pattern, i.e. such that for every $x$, there is an $f(x) = f(x + nT)$, where $T$ is the period and $n$ is any integer.
### Circular motion
Please see [SL Physics 1#6.1 - Circular motion](/g11/sph3u7/#61-circular-motion) and its subcategory "Angular thingies" for more information.
## Partial function decomposition (PFD)
In order to PFD:
1. Factor the denominator into irreducibly quadratic or linear terms.
2. For each factor, create a term. Where capital letters below are constants:
- A linear factor $Bx+C$ has a term $\frac{A}{Bx+C}$.
- A quadratic factor $Dx^2+Ex+G$ has a term $\frac{H}{Dx^2+Ex+G}$.
3. Set the two equal to each other such that the denominators can be factored out.
4. Create systems of equations to solve for each constant.
