Graphs of Basic Functions and Their Properties

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Section 2.6: Graphs of Basic Functions

Continuity

Continuity is a fundamental concept in mathematics that describes whether a function can be drawn without lifting your pencil from the paper. A function is continuous over an interval if its graph has no breaks, holes, or jumps within that interval.

Continuous Function: A function is continuous on an interval if you can draw its graph over that interval without lifting your pencil.
Discontinuity: A point where the function is not continuous (e.g., a hole or jump in the graph).

Example: Determining intervals of continuity for a function. If a function is continuous everywhere except at a point (e.g., x = 3), then it is continuous on the intervals to the left and right of that point.

Graph showing a function with a discontinuity at x=3 Graph showing intervals of continuity

Basic Functions and Their Graphs

Identity Function

The identity function is defined as . It is a straight line passing through the origin with a slope of 1.

Domain:
Range:
Continuity: Continuous everywhere
Monotonicity: Increasing on its entire domain

Graph of the identity function f(x) = x

Squaring Function

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

Domain:
Range:
Continuity: Continuous everywhere
Monotonicity: Decreasing on , increasing on

Graph of the squaring function f(x) = x^2

Cubing Function

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

Domain:
Range:
Continuity: Continuous everywhere
Monotonicity: Increasing on its entire domain

Graph of the cubing function f(x) = x^3

Square Root Function

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

Domain:
Range:
Continuity: Continuous on
Monotonicity: Increasing on its entire domain

Graph of the square root function f(x) = sqrt(x)

Cube Root Function

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

Domain:
Range:
Continuity: Continuous everywhere
Monotonicity: Increasing on its entire domain

Graph of the cube root function f(x) = cube root of x

Absolute Value Function

The absolute value function is defined as . Its graph forms a "V" shape with the vertex at the origin.

Domain:
Range:
Continuity: Continuous everywhere
Monotonicity: Decreasing on , increasing on

Graph of the absolute value function f(x) = |x|

Piecewise-Defined Functions

Definition and Graphing

A piecewise-defined function is a function defined by different expressions for different intervals of the domain. To graph such a function, graph each piece on its respective interval.

Each "piece" of the function applies to a specific part of the domain.
Endpoints may be open or closed circles, depending on whether the interval is inclusive or exclusive.

Example: Graphing a piecewise-defined function:

Graph of a piecewise-defined function with two pieces

Greatest Integer Function (Step Function)

Definition

The greatest integer function, denoted , assigns to each real number the greatest integer less than or equal to . This function is also known as the "floor" function.

Domain:
Range: All integers ()
Continuity: Discontinuous at all integer values
Behavior: Constant on each interval for integer

Graph of the greatest integer function f(x) = floor(x)

Example: Table of Values for a Greatest Integer Function

The following table shows sample values for :

x	0	1/2	1	3/2	2	3	4	-1	-2	-3
y	1	1	1	1	2	2	3	0	0	-1

Table of values for a greatest integer function Graph of y = floor(1/2 x + 1)

Application: Step Function in Real Life

Step functions are often used to model situations where quantities change in jumps rather than smoothly. For example, a shipping company charges a base rate for the first 2 pounds and an additional fee for each extra pound or fraction thereof. The cost function is a step function.

For in , cost is $25$ dollars.
For in , cost is $28$ dollars, and so on.

Graph of a step function modeling shipping costs

The Relation x = y2

Functions vs. Relations

A relation is any set of ordered pairs. A function is a special type of relation where each input (domain value) is paired with exactly one output (range value). The equation is a relation but not a function, because for some values of , there are two corresponding values of (one positive and one negative).

Domain:
Range:

Graph of the relation x = y^2

x	0	1	4	9
y	0	\pm 1	\pm 2	\pm 3

Table of values for the relation x = y^2

Note: The vertical line test can be used to determine if a relation is a function. If any vertical line crosses the graph more than once, the relation is not a function.