Functions and Their Properties

Definition of a Function

A function is a rule that assigns to each element in a set called the domain exactly one element in a set called the range.

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

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

To determine if an equation represents a function, check if each input value corresponds to exactly one output value. If y can be written explicitly as a function of x, it is a function.

Function Notation

• f(x): Read as "f of x," represents the output when x is the input.

• Example: If , then .

Finding Domain and Range

• Given a set of points, the domain is the set of all x-values, and the range is the set of all y-values.

• Restrictions on the domain include:

  • No negative numbers under an even root (e.g., square root).

  • No division by zero.

  • Special restrictions for logarithmic and piecewise functions.

Vertical Line Test

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

Intercepts

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

• x-intercept: Set y = 0 (or f(x) = 0) and solve for x.

• Example: For , the y-intercept is , and the x-intercept is .

More About Functions

Difference Quotient

The difference quotient is used to compute the average rate of change of a function:

• Example: If , then .

Piecewise Functions

A piecewise function is defined by different expressions depending on the input value.

• Example:

• To evaluate, use the rule corresponding to the input value.

Absolute Maximum and Minimum

• Absolute maximum: is the largest value on an interval.

• Absolute minimum: is the smallest value on an interval.

• The location of the max/min is given by the input value c.

Increasing and Decreasing Functions

• A function is increasing on an interval if, for any , .

• A function is decreasing on an interval if, for any , .

Even and Odd Functions

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

• Odd function: for all in the domain. Graph is symmetric about the origin.

Step Functions

• Also called greatest integer functions or floor functions.

• Notation: or , which gives the greatest integer less than or equal to .

Linear Functions and Equations

General Form of a Line

The general form is , where , , and are real numbers.

• y-intercept: Set and solve for .

• x-intercept: Set and solve for .

Special Lines

• Horizontal lines: (slope )

• Vertical lines: (undefined slope)

Slope of a Line

• Slope formula:

• Horizontal lines have zero slope; vertical lines have undefined slope.

Point-Slope and Slope-Intercept Forms

• Point-slope form:

• Slope-intercept form:

Applications

• Example: A phone company charges y = 0.02x + 25$.

• To find the bill for 500 minutes: .

• To find minutes for a minutes.

More on Slope: Parallel and Perpendicular Lines

• Parallel lines: Same slope ().

• Perpendicular lines: Slopes are negative reciprocals ().

Average Rate of Change

The average rate of change of from to is:

This is the slope of the secant line between and .

Transformations of Functions

Basic Functions to Know

• (quadratic)

• (cubic)

• (absolute value)

• (square root)

• (identity)

Translations and Transformations

• Vertical shifts: shifts up, shifts down.

• Horizontal shifts: shifts left, shifts right.

• Reflections: reflects over the x-axis; reflects over the y-axis.

• Vertical stretch/shrink: stretches if , shrinks if .

• Horizontal stretch/shrink: shrinks if , stretches if .

Order of Transformations

1. Put the function in graphing form.

2. Do horizontal shifts and stretches/shrinks first.

3. Do reflections next.

4. Do vertical shifts last.

Combinations and Composition of Functions

Operations with Functions

• Addition:

• Subtraction:

• Multiplication:

• Division: ,

Composition of Functions

• To find the domain, compose the functions and determine where the result is defined.

Decomposition

• Breaking a function into simpler functions whose composition gives the original function.

Identity Function

• The identity function is .

Inverse Functions

Definition

• Two functions and are inverses if for all in their domains.

• Only one-to-one functions have inverses.

• To find the inverse:

  1. Rewrite as .

  2. Switch and .

  3. Solve for .

  4. Rewrite as .

• Inverse functions switch the roles of domain and range.

Distance and Midpoint Formulas

Midpoint Formula

Given two points and , the midpoint is:

Distance Formula

The distance between and is: