3.1 Functions and Their Properties

Definition of a Function

A function is a relation in which each element of the domain (input) is paired with exactly one element of the range (output). If a domain value (x) is used more than once for different outputs, the relation is not a function.

• Example: f(x) = x^2 is a function; x = ±√y is not a function because a single y-value can correspond to two x-values.

• Domain: Set of all possible input values (x-values).

• Range: Set of all possible output values (y-values).

Difference Quotient

The difference quotient is used to compute the average rate of change of a function over an interval and is foundational for calculus.

• Formula: , where

• Steps:

  1. Replace x with (x + h) in the function.

  2. Subtract f(x).

  3. Simplify the numerator.

  4. Factor and cancel h if possible.

• Example: For ,

Domain of a Function

To find the domain, identify all x-values for which the function is defined.

• Linear equations: Domain is all real numbers ().

• Quadratic equations: Domain is all real numbers ().

• Rational functions: Exclude x-values that make the denominator zero.

• Square root functions: The radicand must be non-negative ().

• Example: , domain is .

• Example: , domain is .

3.2 Graphs of Functions

Vertical Line Test

A graph represents a function if and only if no vertical line intersects the graph more than once.

• If a vertical line crosses the graph more than once, the relation is not a function.

Intercepts

• x-intercept: Set y = 0 and solve for x.

• y-intercept: Set x = 0 and solve for y.

• Example: For , x-intercepts at ; y-intercept at .

Symmetry

• Even function: Symmetric about the y-axis.

• Odd function: Symmetric about the origin.

Finding Points and Intercepts

• To find a point, plug in a value for x and solve for y.

• To find intercepts, set the other variable to zero and solve.

3.3 Even and Odd Functions

Functions can be classified as even, odd, or neither based on their symmetry properties.

• Even Function: (symmetric about the y-axis)

• Odd Function: (symmetric about the origin)

• Example: is even; is odd.

3.4 Types of Functions and Piecewise Functions

Common Functions

• Linear Function:

• Quadratic Function:

• Cubic Function:

• Square Root Function:

• Reciprocal Function:

• Absolute Value Function:

• Greatest Integer Function:

Piecewise Defined Functions

A piecewise function is defined by different expressions over different intervals of the domain.

• Specify the function rule for each interval.

• To find intercepts, plug in values as appropriate for each piece.

3.5 Transformations of Functions

Vertical and Horizontal Shifts

• Vertical Shift: shifts up if , down if .

• Horizontal Shift: shifts left if , right if .

Reflections

• Across x-axis:

• Across y-axis:

Stretches and Compressions

• Vertical Stretch/Compression: stretches if , compresses if .

• Horizontal Stretch/Compression: compresses if , stretches if .

3.16 Geometry Applications

Distance Formula

• Distance between and :

Area and Perimeter

• Rectangle: Area = , Perimeter =

• Triangle: Area =

• Circle: Area = , Circumference =

Pythagorean Theorem

• for right triangles

4.4 Revenue and Maximization

Revenue Function

• Revenue: where x = quantity sold, p = price per unit

• To maximize revenue, express R as a quadratic function and use the vertex formula.

• Vertex Formula: For , the maximum (or minimum) occurs at

• Example: If , expand to and find the maximum using the vertex formula.

Standard Form of Quadratic

• Expand and simplify expressions to write quadratics in standard form:

• Example: