Functions and Their Properties

Definition of a Function

A function is a rule that assigns to each value x in a set D a unique value denoted f(x). The set D is the domain of the function. The range is the set of all values of f(x) produced as x varies over the entire domain.

• Domain: The set of all possible input values (independent variable, e.g., time).

• Range: The set of all possible output values (dependent variable, e.g., height of a moving object).

Example: For the function f(x) = x^2, the domain is all real numbers, and the range is all non-negative real numbers.

Vertical Line Test

A graph represents a function if and only if it passes the vertical line test: every vertical line intersects the graph at most once. If a vertical line intersects the graph more than once, it does not represent a function.

• Practice: Parabolas and lines pass the vertical line test; circles do not.

Composite Functions

Given two functions f and g, the composite function f ˆ g is defined by (f ˆ g)(x) = f(g(x)). It is evaluated in two steps: first, apply g to x, then apply f to the result.

• Domain of Composite: All x in the domain of g such that g(x) is in the domain of f.

• Example: If f(x) = 3x^2 - x and g(x) = 1/x, then f(g(x)) = 3(1/x)^2 - (1/x).

Difference Quotient and Secant Line

Secant Line and Difference Quotient

The line between two points P(x, f(x)) and Q(x + h, f(x + h)) on the curve is called the secant line. The slope of the secant line is known as the difference quotient:

• Alternatively, given points P(a, f(a)) and Q(x, f(x)):

Example: For f(x) = x^3, the difference quotient is .

Symmetry in Functions

Types of Symmetry

• Symmetric with respect to the y-axis: If (x, y) is on the graph, so is (-x, y). The graph is unchanged when reflected across the y-axis.

• Symmetric with respect to the x-axis: If (x, y) is on the graph, so is (x, -y). The graph is unchanged when reflected across the x-axis.

• Symmetric with respect to the origin: If (x, y) is on the graph, so is (-x, -y). The graph is unchanged when reflected across the origin.

Even and Odd Functions

• Even function: for all x in the domain. The graph is symmetric about the y-axis.

• Odd function: for all x in the domain. The graph is symmetric about the origin.

Example: f(x) = x^4 - 2x^2 - 20 is even; h(x) = 1/(x^3 - x) is odd.

Categories of Families of Functions

• Domain of polynomials: All real numbers.

• Domain of rational functions: All real numbers except where the denominator is zero.

• Domain of exponential functions: All real numbers.

• Domain of logarithmic functions: Positive real numbers only.

Example: (polynomial), (rational), (exponential), (logarithmic).

Linear, Piecewise, Power, and Root Functions

Linear Functions

• General form:

• Point-slope form:

• Example: Find the linear function for a line passing through (2, 1) with slope 2:

Piecewise Functions

• Defined by different expressions on different intervals of the domain.

• Example:

Power and Root Functions

• Power function: , where n is a positive integer.

• Root function: , where n is a positive integer.

• Even power functions are symmetric about the y-axis; odd power functions are symmetric about the origin.

Slope and Area Functions

Slope Function

The slope function, S(x), is the slope of the curve y = f(x) at the point (x, f(x)).

• Example: For f(x) = x, S(x) = 1; for f(x) = x^2, S(x) = 2x.

Area Function

The area function, A(x), is the area of the region bounded by the graph f(x), the x-axis, and the vertical line x.

• Example: For f(x) = x from x = 0 to x = a,

Summary Table: Families of Functions

Additional info: These notes cover foundational concepts in functions, including definitions, properties, symmetry, and categories, which are essential for further study in calculus.