Functions - Essential Mathematics for Chemical Engineers

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

$Unique Value$

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
$f(x) = x^2$
(1)
A table
$x$ $f$
0 0
1 1
2 4
A graph:
$x vs f(x) Graph$
A description
“Square x.”

$x$	$f$
0	0
1	1
2	4

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”:

Common functions¶

Linear functions (lines)
$y = mx + b$
(2)
where m is the slope and b is the y-intercept.
Power functions (“power law”)
$y = kx^a$
(3)
where k is the prefactor and a is the power.
Polynomials
$y = a_n x^n + \ldots + a_1x + a_0$
(4)
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
$y = a^x$
(5)
where a is the base ( $a > 0$ , $a \ne 1$ ). A special case is $a = e$ , Euler’s number (2.718...).
Logarithmic
$y = \log_a x$
(6)
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
$\sin x \qquad \csc x\\ \cos x \qquad \sec x\\ \tan x \qquad \cot x$
(7)

Manipulating functions¶

Horizontal shift

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

$Shifting in X direction$

Vertical shift

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

$Shifting in Y direction$

Horizontal scaling

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

$Horizontal Compression$

Vertical scaling

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

$Vertical Stretch$

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.

$Unit Circle$

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:

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

(8)

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}

(9)

Each quadrant of the unit circle corresponds to a positive trigonometric function: All, Sine, Tangent, and Cosine.

Special angles¶

30-60-90 triangle

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

$30 60 90 triangle$

45-45-90 triangle

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

$45 45 90 triangle$

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.