Graphs of Basic Functions & Piecewise Functions

Identity, Squaring, and Cubing Functions

The study of basic functions is fundamental in College Algebra. Understanding their graphs, domains, and ranges provides a foundation for more advanced topics.

• Identity Function: Defined as . The graph is a straight line passing through the origin with a slope of 1.

• Domain:

• Range:

• Behavior: Increasing on its entire domain.

• Example: For , respectively.

• Squaring Function: Defined as . The graph is a parabola opening upwards.

• Domain:

• Range:

• Behavior: Decreases on and increases on .

• Example: For , respectively.

• Cubing Function: Defined as . The graph passes through the origin and is symmetric about it.

• Domain:

• Range:

• Behavior: Increasing on its entire domain.

• Example: For , respectively.

Square Root and Cube Root Functions

Root functions are important for understanding transformations and solving equations. Their domains and ranges are more restricted than polynomial functions.

• Square Root Function: Defined as . The graph starts at the origin and increases slowly.

• Domain:

• Range:

• Behavior: Increasing on .

• Example: For , respectively.

• Cube Root Function: Defined as . The graph passes through the origin and is symmetric about it.

• Domain:

• Range:

• Behavior: Increasing on its entire domain.

• Example: For , respectively.

Absolute Value Function

The absolute value function is a classic example of a piecewise-defined function. It is used to measure distance from zero and is fundamental in solving inequalities.

• Definition:

• Domain:

• Range:

• Behavior: Decreases on and increases on .

• Example: For , respectively.

Piecewise-Defined Functions

Piecewise-defined functions are functions that use different expressions for different intervals of their domain. They are useful for modeling situations where a rule changes based on input value.

• Example:

• To evaluate: Choose the correct expression based on the value of .

• Example Calculation: For , use : .

Graphing Linear Functions

Linear functions are the simplest type of function and are foundational for understanding more complex functions. Their graphs are straight lines.

• Example:

• Table of values: For , respectively.

• Graph: A straight line with slope 1 and y-intercept 2.

• Example:

• Table of values: For , respectively.

• Graph: A straight line with slope 1 and y-intercept -3.

Summary Table: Basic Functions

The following table summarizes the properties of the basic functions discussed above.