Overview of Common Functions and Their Graphs in College Algebra

Common Functions and Their Most Basic Graphs

This section provides an overview of the most frequently encountered functions in College Algebra, including their standard forms, basic graphs, and key 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 a parabola 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., $n=1,3,5$): - $f(x) = \frac{1}{x}$, $f(x) = \frac{1}{x^3}$, $f(x) = \frac{1}{x^5}$ - Graph: Hyperbola with branches in opposite quadrants. - Domain: $x \neq 0$ - Range: $y \neq 0$

• For even $n$ (e.g., $n=2,4,6$): - $f(x) = \frac{1}{x^2}$, $f(x) = \frac{1}{x^4}$, $f(x) = \frac{1}{x^6}$ - 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 $x$. The form is $f(x) = x^{1/n}$ or $f(x) = \sqrt[n]{x}$.

• For odd $n$ (e.g., $n=3$): - $f(x) = \sqrt[3]{x}$ - Graph: Passes through the origin, defined for all real $x$.

• For even $n$ (e.g., $n=2$): - $f(x) = \sqrt{x}$ - Graph: Starts at the origin and increases to the right. - Domain: $[0, \infty)$ - Range: $[0, \infty)$

Absolute Value Functions

The absolute value function is defined as $f(x) = |x|$. It outputs the non-negative value of $x$.

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

• Domain: $\mathbb{R}$

• Range: $[0, \infty)$

Piecewise-Defined Functions

Piecewise-defined functions use different formulas for different parts of their domain. Each formula applies to a specific interval of $x$.

• 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
Reciprocal	$f(x) = \frac{1}{x}$	$x \neq 0$	$y \neq 0$	Hyperbola
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$	Half-parabola
Absolute Value	$f(x) = |x|$	$\mathbb{R}$	$[0, \infty)$	V-shape

Worked Examples

• 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 each of the given $x$-values. $f(x) = \begin{cases} \frac{2}{3}x - 11 & \text{if } x \leq -3 \\ -2x^2 - x & \text{if } -3 < x \leq -1 \\ \frac{1}{3}x + 5 & \text{if } x > -1 \end{cases}$ a) $f(-3)$ b) $f(0)$ c) $f(9)$ For each value, determine which case applies and substitute $x$ to find $f(x)$.

• Example 3: Evaluate the piecewise function at each of the given $x$-values. $f(x) = \begin{cases} x^2 - 1 & \text{if } x < 1 \\ 3 & \text{if } x = 1 \\ -2x + 5 & \text{if } x > 1 \end{cases}$ a) $f(0)$ b) $f(1)$ c) $f(4)$ Follow the same process as above.

• Example 4: Graph the piecewise function given. $f(x) = \begin{cases} 2x + 2 & \text{if } x < 3 \\ x^2 - 2 & \text{if } x \geq 3 \end{cases}$ Graph each piece on its respective interval, ensuring continuity or noting any jumps.

Summary

• Recognizing the basic forms and graphs of common functions is crucial for success in College Algebra.

• Pay attention to the domain and range of each function type.

• Piecewise functions require careful attention to the intervals for which each formula applies.