Types of Functions

Basic Function Types

Understanding the basic types of functions is fundamental in College Algebra. Functions can be classified by their algebraic form and graphical behavior.

• Linear Functions: Functions of the form where and are constants. Their graphs are straight lines.

• Quadratic Functions: Functions of the form . Their graphs are parabolas.

• Square Root Functions: Functions of the form or .

• Absolute Value Functions: Functions of the form .

• Cubic Functions: Functions of the form .

• Piecewise Functions: Functions defined by different expressions for different intervals of the domain.

Example: The function is a square root function shifted left by 3 units.

Example: The function is a quadratic function reflected across the x-axis and shifted up by 3 units.

Piecewise Functions

Piecewise functions are defined by different expressions depending on the value of the input variable.

• Definition: A piecewise function uses multiple sub-functions, each with its own domain.

• Example: for , for .

Example: for , for , for .

Greatest Integer Function

The greatest integer function, also known as the floor function, returns the largest integer less than or equal to a given number.

• Notation:

• Graph: The graph is a step function, with jumps at integer values.

• Domain: All real numbers.

• Range: All integers.

Analyzing Graphs of Functions

Zeros of Functions

The zero of a function is the input value where the output is zero. Graphically, zeros correspond to x-intercepts.

• Definition:

• Example: For , the zero is .

Example: For , the zeros are and .

Example: For , the zero is .

Example: For , there are no real zeros.

Intervals of Increase, Decrease, and Constancy

A function may be increasing, decreasing, or constant over different intervals of its domain.

• Increasing: The function's output rises as the input increases.

• Decreasing: The function's output falls as the input increases.

• Constant: The function's output remains unchanged as the input increases.

Example: The graph below shows intervals of increase, decrease, and constancy.

Algebra of Functions

Operations on Functions

Functions can be combined using addition, subtraction, multiplication, and division.

• Sum:

• Difference:

• Product:

• Quotient: ,

Domain: The domain of the combined function is the intersection of the domains of and , except for division where .

Composition of Functions

Composite Functions

The composition of two functions and is written as . The domain of the composite function is the set of all in the domain of $g$ for which is in the domain of $f$.

• Example: If and , then .

Symmetry of Functions

Algebraic Tests of Symmetry

Symmetry helps classify functions and their graphs.

• x-axis Symmetry: Replace with ; if the equation is unchanged, the graph is symmetric about the x-axis.

• y-axis Symmetry: Replace with ; if the equation is unchanged, the graph is symmetric about the y-axis.

• Origin Symmetry: Replace with and with ; if the equation is unchanged, the graph is symmetric about the origin.

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

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

Transformations of Functions

Vertical and Horizontal Translations

Translations shift the graph of a function up, down, left, or right.

• Vertical Translation: shifts up by units; shifts down by $b$ units.

• Horizontal Translation: shifts right by units; shifts left by $d$ units.

Example: Comparing and shows a leftward shift by 2 units.

Reflections

Reflections flip the graph across the x-axis or y-axis.

• x-axis Reflection:

• y-axis Reflection:

Example: The graph of is the reflection of across the x-axis.

Example: The graph of is the reflection of across the y-axis.

Vertical and Horizontal Stretching/Shrinking

Stretching and shrinking change the shape of the graph by scaling it vertically or horizontally.

• Vertical Stretch: , stretches vertically.

• Vertical Shrink: , shrinks vertically.

• Horizontal Stretch: , stretches horizontally.

• Horizontal Shrink: , shrinks horizontally.

Example: The graph below shows vertical stretching and shrinking.

Example: The graph below shows horizontal shrinking.

Example: The graph below shows horizontal stretching.

Example: The graph below shows horizontal stretching with reflection across the y-axis.

Applications of Functions

Functions are used to model real-world situations, such as distance, area, and other quantities.

• Example: The distance between two cars moving at right angles can be modeled using the Pythagorean Theorem: .

Additional info: The domain of this function is , since time cannot be negative.