# Functions

A function is a rule that assigns a unique value $f(x)$ to an input $x$.

```{image} ./_images/x_to_f(x).jpg
:alt: Unique Value
:width: 300px
:align: center
```

We call *x* the *independent variable* and $y = f(x)$ the *dependent variable*.
The *domain* of *f* is all possible values of *x*, while the *range* of *f* is
all possible values of *y*.

A function can be represented by:

- An equation

  $$f(x) = x^2$$

- A table

  | $x$ | $f$ |
  |-----|-----|
  | 0   |  0  |
  | 1   |  1  |
  | 2   |  4  |

- A graph:

  ```{image} ./_images/quadratic_graph.jpg
  :alt: x vs f(x) Graph
  :width: 150px
  :align: center
  ```

- A description

  > "Square x."

Not all curves are functions! *f* must uniquely map *x*. For example, the
circle $x^2 + y^2 = 4$, or equivalently, $y = \pm\sqrt{4-x^2}$ is not a function
because one value of *x* can map to multiple values of *y*. Graphically,
this fails the "vertical line test":

```{topic} Vertical line test
A vertical line cannot intersect a function twice.
```

## Common functions

- Linear functions (lines)

  $$y = mx + b$$

  where *m* is the slope and *b* is the *y*-intercept.

- Power functions ("power law")

  $$y = kx^a$$

  where *k* is the prefactor and *a* is the power.

- Polynomials

  $$y = a_n x^n + \ldots + a_1x + a_0$$

  where *n* is the degree and $a_i$ is the coefficient of the *i*-th power of
  *x*. The polynomial is quadratic when $n=2$, cubic when $n=3$, and quartic
  when $n=4$.

- Exponential

  $$y = a^x$$

  where *a* is the base ($a > 0$, $a \ne 1$). A special case is $a = e$,
  Euler's number (2.718...).

- Logarithmic

  $$y = \log_a x$$

  where *a* is the base ($a > 0$, $a \neq 1$). Two special cases are $a = e$,
  which we call the *natural logarithm* $\ln$, and $a = 10$, which we call the
  *common logarithm*, $\log_{10}$.

- Trigonometric

  $$
  \sin x \qquad \csc x\\
  \cos x \qquad \sec x\\
  \tan x \qquad \cot x
  $$

## Manipulating functions

::::{grid}
:gutter: 2

:::{grid-item-card} Horizontal shift
:columns: 6

- $f(x+a)$ shifts *f* left by *a*.
- $f(x-a)$ shifts *f* right by *a*.

```{image} ./_images/Shifting_X_direction.jpg
:alt: Shifting in X direction
:width: 300px
:align: center
```

:::

:::{grid-item-card} Vertical shift
:columns: 6

- $f(x)+b$ shifts *f* up by *b*.
- $f(x)-b$ shifts *f* down by *b*.

```{image} ./_images/Shifting_Y_direction.jpg
:alt: Shifting in Y direction
:width: 300px
:align: center
```

:::

:::{grid-item-card} Horizontal scaling
:columns: 6

- Scaling $f(cx)$ compresses *f* horizontally by *c*.
- Scaling $f(x/c)$ stretches *f* horizontally by *c*.

```{image} ./_images/Horizontal_Compression.jpg
:alt: Horizontal Compression
:width: 300px
:align: center
```

:::

:::{grid-item-card} Vertical scaling
:columns: 6

- Scaling $f(x)/c$ compresses *f* vertically by *c*.
- Scaling $cf(x)$ streches *f* vertically by *c*.

```{image} ./_images/Vertical_Stretch.jpg
:alt: Vertical Stretch
:width: 300px
:align: center
```

:::
::::

## Trigonometric functions

The given unit circle has a triangle drawn within which represents how
trigonometric functions are related to lengths and heights of the sides of right
triangles.

```{image} ./_images/MAtappunitcircle66.png
:alt: Unit Circle
:width: 300px
:align: center
```

The circle has a radius and is centered at (0,0). A point on the circle is
defined by angle $\theta$ measured counterclockwise from the positive *x*-axis.
Angles can be measured in degrees or radians:

\begin{equation}
\theta({\rm rad.}) = \theta({\rm deg.}) \left( \frac{\pi}{180^{\circ}} \right)
\end{equation}

The trigonometric functions describe how the coordinates of a point on the unit
circle relate to the angle $\theta$:

\begin{align}
\sin(\theta)& = \frac{y}{r}&
\cos(\theta)& = \frac{x}{r}&
\tan(\theta)& = \frac{y}{x}\\
\csc(\theta)& = \frac{r}{y}&
\sec(\theta)& = \frac{r}{x}&
\cot(\theta)& = \frac{x}{y}\\
\end{align}

Each quadrant of the unit circle corresponds to a positive trigonometric
function: **A**ll, **S**ine, **T**angent, and **C**osine.

```{tip}
A mnemonic to remember this order is "All Students Take Calculus".
```

### Special angles

::::{grid}

:::{grid-item-card} 30-60-90 triangle

A right triangle with $30^\circ$ ($\pi/6$) and $60^\circ$ ($\pi/3$) angles
has the following side lengths:

```{image} ./_images/306090triangle.png
:alt: 30 60 90 triangle
:width: 300px
:align: center
```

:::

:::{grid-item-card} 45-45-90 triangle

A right triangle with $45^\circ$ ($\pi/4$) angles has the following side lengths:

```{image} ./_images/454590triangle.png
:alt: 45 45 90 triangle
:width: 300px
:align: center
```

:::
::::

and so

| $\theta$  |         | $\sin \theta$  | $\cos \theta$ |
|------------|---------|----------------|---------------|
| $30^\circ$ | $\pi/6$ | $1/2$          | $\sqrt{3}/2$  |
| $45^\circ$ | $\pi/4$ | $1/\sqrt{2}$   | $1/\sqrt{2}$  |
| $60^\circ$ | $\pi/3$ | $\sqrt{3}/2$   | $1/2$         |

What about angles greater than $90^\circ$? Take the angle of $\theta$ relative
to the *x*-axis, then apply the sign based on which quadrant the angle falls in.

```{image} ./_images/supplementary_angles.png
:width: 300px
:align: center
```

For example, if $\theta = 135^\circ = 3\pi/4$, then the angle relative to the
*x*-axis is $\theta' = 45^\circ = \pi/4$. Hence,

\begin{align}
\sin \theta' &= \frac{1}{\sqrt{2}} & \cos \theta' &= \frac{1}{\sqrt{2}} \\
\sin \theta &= \frac{1}{\sqrt{2}} & \cos \theta &= -\frac{1}{\sqrt{2}}
\end{align}

### Physical meaning

In addition to their geometric significance, trigonometric functions typically
represent physical waves.

```{image} ./_images/Triggraph.png
:alt: Trigonometric Function
:width: 300px
:align: center
```

The properties of the wave can be encoded by manipulating the function.

\begin{equation}
y = A\sin\left[ \frac{2\pi}{L}(x + x_0) \right] + y_0
\end{equation}

where *A* is the amplitude of the wave, *L* is the period of the wave, $x_0$ is
a horizontal (phase) shift, and $y_0$ is a vertical shift.