Functions and Their Graphs: College Algebra Study Notes

1.1 Functions

Functions in the World Around Us

Functions are mathematical rules that describe relationships between quantities in various real-world contexts, such as physics, economics, and biology.

• Key Point: Many phenomena can be modeled using functions, allowing us to predict and analyze outcomes.

• Example: The conversion between Celsius and Fahrenheit temperatures is a function.

Definition of a Function

A function is a rule that assigns to each element x in a set A exactly one element, denoted f(x), in a set B.

• Key Point: Functions are often represented by letters such as f, g, or h.

• Example: If f is the rule "square the number," then f(x) = x^2.

Ways to Represent a Function

Functions can be represented in multiple ways to suit different contexts and applications.

• Verbal: Describing the rule in words (e.g., "Multiply the Celsius temperature by 9/5, then add 32").

• Algebraic: Using a formula (e.g., $A(r) = \pi r^2$ for the area of a circle).

• Graphical: Using a graph to show the relationship between variables.

• Numerical: Using a table of values.

Representation	Example
Verbal	Convert Celsius to Fahrenheit: multiply by 9/5, add 32
Algebraic	$A(r) = \pi r^2$
Graphical	Graph of earthquake acceleration over time
Numerical	Table of postage costs by weight

1.2 Graphs of Functions

Plotting Functions by Points

Graphing a function involves plotting points that satisfy the function's rule and connecting them to visualize the relationship.

• Linear Functions: Functions of the form $f(x) = mx + b$ are called linear functions. Their graphs are straight lines with slope m and y-intercept b.

• Constant Functions: If $m = 0$, the function $f(x) = b$ is a constant function, represented by a horizontal line $y = b$.

• Example: The graph of $y = 2x + 1$ is a straight line with slope 2 and y-intercept 1.

Power and Root Functions

Functions of the form $f(x) = x^n$ are called power functions, and those of the form $f(x) = x^{1/n}$ are root functions.

• Key Point: Power functions include quadratic ($f(x) = x^2$), cubic ($f(x) = x^3$), etc. Root functions include square root ($f(x) = \sqrt{x}$), cube root ($f(x) = \sqrt[3]{x}$), etc.

• Example: The graph of $f(x) = x^2$ is a parabola opening upwards.

Piecewise Defined Functions

Some functions are defined by different rules for different parts of their domain. These are called piecewise defined functions.

• Key Point: Piecewise functions are useful for modeling situations where a rule changes based on input value.

• Example: A tax rate that changes at different income levels.

Vertical Line Test: Which Graphs Represent Functions?

The vertical line test is a graphical method to determine if a curve represents a function.

• Key Point: If any vertical line intersects the graph more than once, the graph does not represent a function.

• Example: The graph of $y = x^2$ passes the vertical line test, but a circle does not.

Extracting Information from the Graph of a Function

The graph of a function provides valuable information about its domain, range, and behavior.

• Domain: The set of all possible input values (x) for which the function is defined.

• Range: The set of all possible output values (f(x)).

• Example: For $f(x) = \sqrt{4 - x^2}$, the domain is $[-2, 2]$ and the range is $[0, 2]$.

Solving Equations and Inequalities Graphically

Equations and inequalities involving functions can be solved by analyzing their graphs.

• Key Point: The solution to $f(x) = g(x)$ is the set of x-values where the graphs of f and g intersect.

• Key Point: The solution to $f(x) < g(x)$ is the set of x-values where the graph of f lies below that of g.

• Example: For $f(x) = 2x^2 + 3$ and $g(x) = 5x + 6$, the solutions to $f(x) = g(x)$ are $x = -0.5$ and $x = 3$.

Increasing and Decreasing Functions

A function is increasing on an interval if its graph rises as x increases, and decreasing if its graph falls.

• Definition:

  • Increasing: $f(x_1) < f(x_2)$ whenever $x_1 < x_2$ in the interval.

  • Decreasing: $f(x_1) > f(x_2)$ whenever $x_1 < x_2$ in the interval.

• Example: For $f(x) = 12x^3 + 4x - 3x^4$, the function increases on $(-\infty, -1)$ and $(0, 2)$, and decreases on $[-1, 0]$ and $[2, \infty)$.

Maximum and Minimum Values of a Function

Functions can have local maximum and local minimum values at certain points in their domain.

• Local Maximum: $f(a)$ is a local maximum if $f(a) \geq f(x)$ for all x near a.

• Local Minimum: $f(a)$ is a local minimum if $f(a) \leq f(x)$ for all x near a.

• Example: For $f(x) = x^3 - 8x + 1$, the graph shows local maxima and minima at specific points.

Additional info: Some context and examples have been inferred and expanded for clarity and completeness.