Overview of Common Functions and Their Graphs in College Algebra

Common Functions & Their Most Basic Graphs

This study guide provides an overview of the most common types of functions encountered in College Algebra, including their basic forms, graphs, and properties. Understanding these foundational functions is essential for analyzing more complex mathematical relationships.

Polynomial Functions

Polynomial functions are expressions of the form $f(x) = x^n$, where $n$ is a non-negative integer. The behavior and graph of the function depend on whether $n$ is odd or even.

• Linear Function ($n=1$): $f(x) = x$ - Graph: Straight line through the origin with a slope of 1. - Domain: All real numbers ($\mathbb{R}$). - Range: All real numbers ($\mathbb{R}$).

• Quadratic Function ($n=2$): $f(x) = x^2$ - Graph: Parabola opening upwards. - Domain: $\mathbb{R}$ - Range: $[0, \infty)$

• Cubic Function ($n=3$): $f(x) = x^3$ - Graph: S-shaped curve passing through the origin. - Domain: $\mathbb{R}$ - Range: $\mathbb{R}$

• Quartic Function ($n=4$): $f(x) = x^4$ - Graph: Similar to quadratic but flatter near the origin. - Domain: $\mathbb{R}$ - Range: $[0, \infty)$

• Quintic Function ($n=5$): $f(x) = x^5$ - Graph: Similar to cubic but steeper. - Domain: $\mathbb{R}$ - Range: $\mathbb{R}$

Rational Functions

Rational functions are of the form $f(x) = \frac{1}{x^n}$, where $n$ is a positive integer. The graph and properties depend on whether $n$ is odd or even.

• For odd $n$ (e.g., $n=1,3,5$): - $f(x) = \frac{1}{x}$, $f(x) = \frac{1}{x^3}$, etc. - Graph: Hyperbola with branches in opposite quadrants. - Domain: $x \neq 0$ - Range: $y \neq 0$

• For even $n$ (e.g., $n=2,4,6$): - $f(x) = \frac{1}{x^2}$, $f(x) = \frac{1}{x^4}$, etc. - Graph: Both branches above the x-axis, approaching zero as $|x|$ increases. - Domain: $x \neq 0$ - Range: $(0, \infty)$

Root Functions

Root functions involve taking the $n$th root of $x$, written as $f(x) = x^{1/n}$ or $f(x) = \sqrt[n]{x}$.

• Odd roots (e.g., cube root): $f(x) = x^{1/3}$ - Graph: Passes through the origin, defined for all real $x$.

• Even roots (e.g., square root): $f(x) = x^{1/2}$ - Graph: Starts at the origin and increases, defined for $x \geq 0$.

Absolute Value Functions

The absolute value function is defined as $f(x) = |x|$. It measures the distance of $x$ from zero on the number line.

• Graph: V-shaped, vertex at the origin.

• Domain: $\mathbb{R}$

• Range: $[0, \infty)$

Piecewise-Defined Functions

Piecewise-defined functions are defined by different formulas over different intervals of the domain. Each formula applies to a specific portion of the real number line.

• To evaluate a piecewise function at a given $x$-value, first determine which interval $x$ falls into, then use the corresponding formula.

• Graphing involves plotting each piece over its specified interval.

Example Table: Common Functions and Their Properties

Function Type	General Form	Domain	Range	Graph Shape
Linear	$f(x) = x$	$\mathbb{R}$	$\mathbb{R}$	Straight line
Quadratic	$f(x) = x^2$	$\mathbb{R}$	$[0, \infty)$	Parabola
Cubic	$f(x) = x^3$	$\mathbb{R}$	$\mathbb{R}$	S-curve
Rational (odd $n$)	$f(x) = \frac{1}{x^n}$	$x \neq 0$	$y \neq 0$	Hyperbola
Rational (even $n$)	$f(x) = \frac{1}{x^n}$	$x \neq 0$	$(0, \infty)$	Both branches above x-axis
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$	Half-parabola
Absolute Value	$f(x) = |x|$	$\mathbb{R}$	$[0, \infty)$	V-shape

Examples and Applications

• Example 1: Graph each function using knowledge of the basic function and plotting key points. Then, determine the function type, domain, and range. a) $f(x) = -\sqrt{x-1}$ b) $f(x) = \frac{1}{2}|x| + 3$ For each, identify the parent function, apply transformations, and state the domain and range.

• Example 2: Evaluate the piecewise function at given $x$-values. For example, if $f(x)$ is defined differently for $x < -3$, $-3 \leq x \leq -1$, and $x > -1$, use the correct formula for each $x$.

• Example 3: Evaluate another piecewise function at specified $x$-values, following the same process.

• Example 4: Graph a piecewise function by plotting each segment over its defined interval.

Additional info: Understanding the basic shapes and properties of these functions is crucial for graphing, solving equations, and modeling real-world situations in algebra.