Graphs of Common Functions: Key Types and Properties

Graphs, Functions, and Models

Common Functions and Their Graphs

This section introduces several fundamental types of functions frequently encountered in Precalculus. Understanding their graphs, domains, and ranges is essential for analyzing more complex functions and mathematical models.

Constant Function

• Definition: A function that always returns the same value, regardless of input.

• General Form: $f(x) = c$ where $c$ is a constant.

• Domain: $(-0, 0)$

• Range: $\{c\}$

• Graph: A horizontal line at $y = c$.

• Example: $f(x) = 2$

Identity Function

• Definition: A function that returns its input unchanged.

• General Form: $f(x) = x$

• Domain: $(-0, 0)$

• Range: $(-0, 0)$

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

• Example: $f(x) = x$

Square Function

• Definition: A function that squares its input.

• General Form: $f(x) = x^2$

• Domain: $(-0, 0)$

• Range: $[0, 0)$

• Graph: A parabola opening upwards with vertex at the origin.

• Example: $f(x) = x^2$

Cube Function

• Definition: A function that cubes its input.

• General Form: $f(x) = x^3$

• Domain: $(-0, 0)$

• Range: $(-0, 0)$

• Graph: An S-shaped curve passing through the origin.

• Example: $f(x) = x^3$

Square Root Function

• Definition: A function that returns the non-negative square root of its input.

• General Form: $f(x) = \sqrt{x}$

• Domain: $[0, 0)$

• Range: $[0, 0)$

• Graph: A curve starting at the origin and increasing slowly to the right.

• Example: $f(x) = \sqrt{x}$

Cube Root Function

• Definition: A function that returns the cube root of its input.

• General Form: $f(x) = \sqrt[3]{x}$

• Domain: $(-0, 0)$

• Range: $(-0, 0)$

• Graph: A curve passing through the origin, increasing slowly for large $|x|$.

• Example: $f(x) = \sqrt[3]{x}$

Comparison Table: Domains and Ranges of Common Functions

Function	General Form	Domain	Range
Constant	$f(x) = c$	$(-0, 0)$	$\{c\}$
Identity	$f(x) = x$	$(-0, 0)$	$(-0, 0)$
Square	$f(x) = x^2$	$(-0, 0)$	$[0, 0)$
Cube	$f(x) = x^3$	$(-0, 0)$	$(-0, 0)$
Square Root	$f(x) = \sqrt{x}$	$[0, 0)$	$[0, 0)$
Cube Root	$f(x) = \sqrt[3]{x}$	$(-0, 0)$	$(-0, 0)$

Additional info: These functions form the basis for understanding more complex function transformations and compositions in Precalculus. Mastery of their properties is essential for graphing, solving equations, and modeling real-world phenomena.