Linear and Nonlinear Functions: Key Concepts in College Algebra

Linear Functions and Constant Functions

Definition and Properties

A linear function is a function represented by the equation $f(x) = mx + b$, where m and b are constants.

• m is the slope (rate of change) of the function.

• b is the y-intercept, the point where the line crosses the y-axis.

Example:

• $f(x) = -5 + 2x$

• $f(x) = \frac{1}{2} - 3x$ → $m = -3$, $b = \frac{1}{2}$

If the value of m is zero, then $f(x) = b$ is a constant function. Its graph is a horizontal line.

• Example: $f(x) = 5$

• Here, $m = 0$ and the y-intercept is $5$.

Slope of a Linear Function

The slope of a line passing through points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$

• Positive slope: line rises from left to right ($m > 0$)

• Negative slope: line falls from left to right ($m < 0$)

• Zero slope: horizontal line ($m = 0$)

• Undefined slope: vertical line (not a function)

Example: Find the slope between $(2,1)$ and $(-3,4)$:

• $m = \frac{1-4}{2-(-3)} = \frac{-3}{5}$

Example: Find the slope between $(4,0)$ and $(2,2)$:

• $m = \frac{0-2}{4-2} = \frac{-2}{2} = -1$

Zeros (x-intercepts) of a Function

The zero or x-intercept of a function is the value of $x$ where $f(x) = 0$.

• If a graph crosses the x-axis at $(-3,0)$, then $-3$ is a zero of the function.

• The y-intercept is the value of $y$ when $x = 0$.

Example: For $f(x) = 2x - 4$:

• Set $f(x) = 0$: $2x - 4 = 0 \implies x = 2$ (x-intercept)

• Set $x = 0$: $f(0) = -4$ (y-intercept)

Nonlinear Functions

Definition and Identification

A nonlinear function is any function whose graph is not a straight line. Nonlinear functions include quadratic, cubic, square root, and absolute value functions.

• Linear functions have a constant rate of change (slope).

• Nonlinear functions have a variable rate of change.

Example: $g(x) = x^2 + 1$ is nonlinear because the output changes at a non-constant rate as $x$ changes.

Table: Linear vs. Nonlinear Functions

Function	Type	Reason
$f(x) = 3x - 4$	Linear	Constant slope
$g(x) = x^2 + 1$	Nonlinear	Variable rate of change

Basic Linear and Nonlinear Functions

Common Functions, Graphs, Domains, and Ranges

• Identity Function: $f(x) = x$ Domain: $(-\infty, \infty)$ Range: $(-\infty, \infty)$

• Square Function: $f(x) = x^2$ Domain: $(-\infty, \infty)$ Range: $[0, \infty)$

• Cube Function: $f(x) = x^3$ Domain: $(-\infty, \infty)$ Range: $(-\infty, \infty)$

• Square Root Function: $f(x) = \sqrt{x}$ Domain: $[0, \infty)$ Range: $[0, \infty)$

• Absolute Value Function: $f(x) = |x|$ Domain: $(-\infty, \infty)$ Range: $[0, \infty)$

Increasing and Decreasing Functions

Definitions and Interval Notation

A function $f$ is:

• Increasing on an interval if, as $x$ increases, $f(x)$ increases.

• Decreasing on an interval if, as $x$ increases, $f(x)$ decreases.

Use interval notation to describe where a function is increasing or decreasing. For example, if a function increases on $(-2, 2)$ and decreases on $(-\infty, -2) \cup (2, \infty)$, we write:

• Increasing: $(-2, 2)$

• Decreasing: $(-\infty, -2) \cup (2, \infty)$

Average Rate of Change

Definition and Formula

The average rate of change of a function $f(x)$ from $a$ to $b$ is:

$\frac{f(b) - f(a)}{b - a}$

This measures the change in $f(x)$ per unit change in $x$ over the interval $[a, b]$.

• Example: If a car's speed is 30 mph at $t=2$ seconds and 24 mph at $t=5$ seconds, the average rate of change from $t=2$ to $t=5$ is $\frac{24-30}{5-2} = \frac{-6}{3} = -2$ mph per second.

Application Example: Torricelli's Law

A cylindrical tank contains 100 gallons of water. The amount of water remaining after $x$ minutes is $A(x) = 100\left(1 - \frac{x}{5}\right)$.

• Calculate the average rate of change of $A$ from $1$ to $1.5$ minutes and from $2$ to $2.5$ minutes.

• Interpret the results and compare the two rates.

Note: The average rate of change may differ over different intervals for nonlinear functions.