Precalculus and Calculus Foundations: Functions, Trigonometry, and Exponential Models

Functions

Definition and Domain of Functions

A function is a rule that assigns to each input exactly one output. The domain of a function is the set of all possible input values (usually x-values) for which the function is defined.

• Finding the Domain: For functions involving square roots, the expression inside the root must be greater than or equal to zero.

• Critical Points: Set the expression inside the root equal to zero to find endpoints of the domain.

• Intervals: Test values in each interval to determine where the function is defined.

Example: The domain of $y = \sqrt{\frac{2-x}{x-3}}$ is $(-\infty, 2] \cup (3, \infty)$.

Recognizing Linear Functions from Tables

A function is linear if the difference in y-values is constant for equal differences in x-values. This can be checked using tabular data.

• If $y$ changes by the same amount as $x$ increases by a fixed amount, the function is linear.

x	0	1	2	3
y	27	25	23	21

x	15	20	25	30
y	62	72	82	92

Example: In both tables above, the change in y is constant for each fixed change in x, indicating a linear relationship.

Trigonometric Functions

Right Triangle Definitions

Trigonometric functions relate the angles of a right triangle to the ratios of its sides.

• Sine: $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

• Cosine: $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

• Tangent: $\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

Special Values of Trigonometric Functions

Certain angles have well-known trigonometric values, often derived from special triangles (30-60-90 and 45-45-90 triangles).

θ in degrees	θ in radians	sin θ	cos θ	tan θ	csc θ	sec θ	cot θ
0°	0	0	1	0	undefined	1	undefined
30°	$\frac{\pi}{6}$	$\frac{1}{2}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{\sqrt{3}}$	2	$\frac{2}{\sqrt{3}}$	$\sqrt{3}$
45°	$\frac{\pi}{4}$	$\frac{\sqrt{2}}{2}$	$\frac{\sqrt{2}}{2}$	1	$\sqrt{2}$	$\sqrt{2}$	1
60°	$\frac{\pi}{3}$	$\frac{\sqrt{3}}{2}$	$\frac{1}{2}$	$\sqrt{3}$	$\frac{2}{\sqrt{3}}$	2	$\frac{1}{\sqrt{3}}$
90°	$\frac{\pi}{2}$	1	0	undefined	1	undefined	0

Graphs of Sine and Cosine

The sine function $y = \sin t$ is an odd function, symmetric about the origin. The cosine function $y = \cos t$ is an even function, symmetric about the y-axis. Both are periodic with period $2\pi$.

• Sine: $y = \sin t$ repeats every $2\pi$ units.

• Cosine: $y = \cos t$ repeats every $2\pi$ units.

Inverse Trigonometric Functions

Definitions and Properties

The inverse trigonometric functions undo the action of the trigonometric functions, but are defined only on restricted domains to ensure they are functions (one-to-one).

Function	Domain	Range
$\sin^{-1}$	$[-1, 1]$	$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$
$\cos^{-1}$	$[-1, 1]$	$[0, \pi]$
$\tan^{-1}$	$\mathbb{R}$	$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

Example: $\sin^{-1}\left(\frac{\sqrt{3}}{2}\right) = \frac{\pi}{3}$

Exponential Functions and Applications

Exponential Growth and Decay

An exponential function has the form $f(x) = ab^x$ or $f(x) = ae^{kx}$, where the rate of change is proportional to the current value. Exponential growth occurs when $b > 1$ or $k > 0$, and decay when $0 < b < 1$ or $k < 0$.

• Doubling Time: The time required for an exponentially growing quantity to double.

• Half-Life: The time required for an exponentially decaying quantity to halve.

Compound Interest

Compound interest is a classic application of exponential growth. The formula for compound interest is:

$A(t) = P \left(1 + \frac{r}{n}\right)^{nt}$

For continuous compounding, the formula is:

$A(t) = Pe^{rt}$

Recognizing Function Types from Tables

Given a table of values, you can classify the function as linear, exponential, or neither by examining the pattern of change:

• Linear: Constant difference between outputs for equal input steps.

• Exponential: Constant ratio between consecutive outputs for equal input steps.

• Power: Output changes by a power of the input.

	1	2	3	4	5	6
F(x)	1	4	9	16	25	36
G(x)	2	4	8	16	32	62
H(x)	1	8	27	64	125	216
K(x)	6	12	24	48	96	186

Example: G(x) doubles each time, indicating an exponential function; F(x) increases by perfect squares, indicating a quadratic (power) function.