Precalculus Study Notes: Functions, Domain & Range, Interval Notation, and More

Functions and Their Representations

Definition of a Function

A function is a relation in which each input results in only one output. This means that for every value of the independent variable (usually x), there is exactly one corresponding value of the dependent variable (usually y).

• Each input can have at most one output. For example, you cannot be in two places at the same time.

• Output values can be used more than once. Multiple inputs can have the same output.

Vertical Line Test: If any vertical line crosses a graph more than once, the graph does not represent a function of x.

Example: The graph of a circle fails the vertical line test and is not a function.

Domain and Range

• Domain: The set of all possible input values (usually x).

• Range: The set of all possible output values (usually y).

Example: For the relation {(1, 3), (2, 1), (4, 3), (4, 0)}, the domain is {1, 2, 4} and the range is {0, 1, 3}.

Function Notation

The special notation f(x), read as "f of x" or "f at x", represents the value of the function at the number x. For example, f(2) means the value of the function when x = 2.

Functions as Equations

If an equation is solved for y and more than one value of y can be obtained for a given x, then the equation does not define y as a function of x.

• Independent variable: Usually x.

• Dependent variable: Usually y, depends on what you put in.

Finding Values of Functions

Evaluating Functions

To find the value of a function for a given input, replace the variable inside the parentheses with the given value and simplify.

• Example: Given f(x) = 3x^2 - 2:

  • Find f(2): $f(2) = 3(2)^2 - 2 = 3(4) - 2 = 12 - 2 = 10$

  • Find f(-1): $f(-1) = 3(-1)^2 - 2 = 3(1) - 2 = 3 - 2 = 1$

Interval Notation

Describing Sets of Numbers

Interval notation is used to describe sets of numbers, especially when expressing domains and ranges.

Interval	Set-Builder Notation	Graph
[a, b]	{x | a ≤ x ≤ b}	Solid line between a and b, closed circles at both ends
(a, b)	{x | a < x < b}	Solid line between a and b, open circles at both ends
[a, b)	{x | a ≤ x < b}	Solid line between a and b, closed circle at a, open at b
(a, b]	{x | a < x ≤ b}	Solid line between a and b, open circle at a, closed at b
(−∞, a)	{x | x < a}	Arrow to the left, open circle at a
(a, ∞)	{x | x > a}	Arrow to the right, open circle at a
(−∞, ∞)	{x | x ∈ ℝ}	Arrow in both directions

• Parentheses indicate endpoints that are not included; square brackets indicate endpoints that are included.

• Order matters: always write the smaller number first.

Domain and Range from Graphs

How to Find Domain and Range

• Domain: Look for the smallest and largest x values on the graph. If the graph extends indefinitely, use interval notation with infinity.

• Range: Look for the smallest and largest y values on the graph.

x-intercepts: Points where the graph crosses the x-axis (set y = 0).

y-intercepts: Points where the graph crosses the y-axis (set x = 0).

Increasing, Decreasing, and Constant Functions

Definitions

• Increasing: The function's output gets larger as x increases.

• Decreasing: The function's output gets smaller as x increases.

• Constant: The function's output stays the same as x increases.

To determine intervals of increase or decrease, look at the y-values as x increases.

Relative Maximum and Minimum

Definitions

• Relative Maximum: The highest point in a particular section of a graph.

• Relative Minimum: The lowest point in a particular section of a graph.

Example: If the point (a, k) is at the top of a hill on the graph of f(x), then f(x) has a relative maximum of k at x = a.

Even and Odd Functions

Definitions

• Even Function: $f(-x) = f(x)$ for all x in the domain. The graph is symmetric with respect to the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all x in the domain. The graph is symmetric with respect to the origin.

Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Difference Quotient

Definition and Formula

The difference quotient is a formula used to compute the average rate of change of a function over an interval:

$\frac{f(x+h) - f(x)}{h}, \quad h \neq 0$

• Find f(x + h), subtract f(x), then divide by h.

Example: Given $f(x) = 2x + 3$, the difference quotient is:

$\frac{2(x+h) + 3 - (2x + 3)}{h} = \frac{2x + 2h + 3 - 2x - 3}{h} = \frac{2h}{h} = 2$

Summary Table: Function Properties

Property	Definition	How to Find
Domain	Set of all possible input values	Look at x-values on graph or in equation
Range	Set of all possible output values	Look at y-values on graph or in equation
x-intercept	Where graph crosses x-axis	Set y = 0 and solve for x
y-intercept	Where graph crosses y-axis	Set x = 0 and solve for y
Increasing	y-values get larger as x increases	Look for upward slope on graph
Decreasing	y-values get smaller as x increases	Look for downward slope on graph
Constant	y-values stay the same as x increases	Look for flat sections on graph
Relative Maximum	Highest point in a section	Look for peaks on graph
Relative Minimum	Lowest point in a section	Look for valleys on graph
Even Function	$f(-x) = f(x)$	Check symmetry about y-axis
Odd Function	$f(-x) = -f(x)$	Check symmetry about origin

Additional info:

• Absolute values of the dependent variable (e.g., $|y|$) are not functions.

• Even powers of the dependent variable (e.g., $y^2$) are not functions.

• When analyzing graphs, always use interval notation for domain and range.

• Polynomial functions with only even exponents are always even functions.