Library of Functions

Constant Function

The constant function is defined as , where b is a real number. This function assigns the same value to every input.

• Domain: All real numbers

• Range: The single value {b}

• Graph: A horizontal line at y = b

• Y-intercept: (0, b)

• Even function: Yes, since

Identity Function

The identity function is defined as . It is a linear function where the output equals the input.

• Domain and Range: All real numbers

• Graph: A line with slope 1 passing through the origin

• Y-intercept: (0, 0)

• Odd function: Yes, since

• Increasing: Over its entire domain

Square Function

The square function is defined as . Its graph is a parabola opening upwards.

• Domain: All real numbers

• Range: All non-negative real numbers ()

• Intercept: (0, 0)

• Even function: Yes, since

• Decreasing: On

• Increasing: On

Cube Function

The cube function is defined as . Its graph is an S-shaped curve passing through the origin.

• Domain and Range: All real numbers

• Intercept: (0, 0)

• Odd function: Yes, since

• Increasing: Over its entire domain

Square Root Function

The square root function is defined as . Its graph starts at the origin and increases slowly.

• Domain: (non-negative real numbers)

• Range:

• Intercept: (0, 0)

• Neither even nor odd: The function does not exhibit symmetry about the y-axis or origin.

• Increasing: On

• Absolute minimum: 0 at

Cube Root Function

The cube root function is defined as . Its graph passes through the origin and is symmetric about the origin.

• Domain and Range: All real numbers

• Intercept: (0, 0)

• Odd function: Yes, since

• Increasing: Over its entire domain

• No absolute extrema: The function does not have absolute maximum or minimum values.

Reciprocal Function

The reciprocal function is defined as . Its graph consists of two branches, one in each quadrant, and has no intercepts.

• Domain: All real numbers except

• Range: All real numbers except

• No intercepts: The graph does not cross the axes

• Odd function: Yes, since

• Decreasing: On and

Absolute Value Function

The absolute value function is defined as . Its graph is a V-shaped curve symmetric about the y-axis.

• Domain: All real numbers

• Range:

• Intercept: (0, 0)

• Even function: Yes, since

• Decreasing: On

• Increasing: On

• Absolute minimum: 0 at

Greatest Integer Function (Step Function)

The greatest integer function, also known as the step function, is defined as , where is the greatest integer less than or equal to x.

• Domain: All real numbers

• Range: All integers

• Y-intercept: (0, 0)

• X-intercepts: In the interval [0, 1)

• Neither even nor odd: The function does not exhibit symmetry

• Constant: On every interval of the form [k, k+1), for integer k

Piecewise-defined Functions

Definition and Properties

A piecewise-defined function is a function defined by different expressions for different intervals of its domain. For example, the absolute value function can be written as:

• if

• if

Piecewise-defined functions are useful for modeling situations where a rule changes depending on the input value.

Example: Application of Piecewise-defined Function

A trucking company charges per pound as follows:

• per mile for the first $100$ miles

• per mile for the next $300$ miles

• per mile for any distance in excess of $400$ miles

If is the charge per pound for hauling miles, then:

• For :

• For :

• For :

This is a classic example of a piecewise-defined function used in real-world applications.

Summary Table: Library of Functions