# MATH 117: Calculus 1

## Functions

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 $y=x$.

<img src="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
    <img src="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 $f(x)=e^x$.
    $$
    \cosh x = \frac{1}{2}(e^x + e^{-x}) \\
    \sinh x = \frac{1}{2}(e^x - e^{-x})
    $$