College Algebra: Functions, Graphs, and Applications Study Guide

Functions and Their Graphs

Increasing, Decreasing, and Constant Intervals

Understanding where a function increases, decreases, or remains constant is fundamental in analyzing its behavior.

• Increasing Interval: A function f(x) is increasing on an interval if, as x increases, f(x) also increases.

• Decreasing Interval: A function f(x) is decreasing on an interval if, as x increases, f(x) decreases.

• Constant Interval: A function f(x) is constant on an interval if, as x increases, f(x) remains the same.

• Example: Given a graph, identify these intervals by observing the slope of the curve.

Local Maxima and Minima

Local maxima and minima are points where the function reaches a highest or lowest value within a certain interval.

• Local Maximum: The highest point in a particular section of a graph.

• Local Minimum: The lowest point in a particular section of a graph.

• Notation: If f(x) has a local maximum at x = a, then f(a) is the local maximum value.

• Example: On a graph, these are the peaks (maxima) and valleys (minima).

Applications of Functions

Modeling with Functions

Functions can be used to model real-world scenarios, such as maximizing area with a fixed perimeter.

• Example: If a fence of length 200 ft is used to enclose a rectangular area, and one side is x ft long, the area A as a function of x is:

$A(x) = x \left(\frac{200 - 2x}{2}\right) = x(100 - x)$

• This quadratic function can be analyzed to find the maximum area.

Piecewise Functions

Definition and Graphing

A piecewise function is defined by different expressions over different intervals of the domain.

• Example:

  • $f(x) = \begin{cases} 2x+1 & x < 1 \\ x^2 & x \geq 1 \end{cases}$

• To graph, plot each piece on its respective interval.

Transformations of Functions

Shifts, Reflections, and Stretches

Transformations alter the position or shape of a function's graph.

• Vertical Shift: $f(x) + k$ shifts the graph up (if k > 0) or down (if k < 0).

• Horizontal Shift: $f(x - h)$ shifts the graph right (if h > 0) or left (if h < 0).

• Reflection: $-f(x)$ reflects over the x-axis; $f(-x)$ reflects over the y-axis.

• Vertical Stretch/Compression: $a f(x)$ stretches if |a| > 1, compresses if 0 < |a| < 1.

• Example: The graph of $y = |x|$ shifted right by 2 units is $y = |x - 2|$.

Difference Quotient

Definition and Calculation

The difference quotient is used to compute the average rate of change of a function over an interval and is foundational in calculus.

• Formula:

  • $\frac{f(x+h) - f(x)}{h}$

• Example: For $f(x) = x^2$, the difference quotient is:

$\frac{(x+h)^2 - x^2}{h} = \frac{x^2 + 2xh + h^2 - x^2}{h} = \frac{2xh + h^2}{h} = 2x + h$

Domain and Range

Finding the Domain

The domain of a function is the set of all possible input values (x-values) for which the function is defined.

• For rational functions: Exclude values that make the denominator zero.

• For square roots: The expression under the root must be non-negative.

• Example: The domain of $f(x) = \frac{1}{x-2}$ is all real numbers except $x = 2$.

Even and Odd Functions

Definitions and Tests

Functions can be classified as even, odd, or neither based on their symmetry.

• Even Function: $f(-x) = f(x)$ for all x in the domain. Graph is symmetric about the y-axis.

• Odd Function: $f(-x) = -f(x)$ for all x in the domain. Graph is symmetric about the origin.

• Neither: If neither condition holds.

• Example: $f(x) = x^2$ is even; $f(x) = x^3$ is odd.

Function Composition and Inverses

Composition

Composing functions involves applying one function to the result of another.

• Notation: $(f \circ g)(x) = f(g(x))$

• Example: If $f(x) = 2x$ and $g(x) = x + 1$, then $(f \circ g)(x) = f(x + 1) = 2(x + 1) = 2x + 2$.

Inverse Functions

• Definition: The inverse function $f^{-1}(x)$ reverses the effect of $f(x)$.

• Finding the Inverse: Swap x and y in the equation and solve for y.

• Example: If $f(x) = 3x + 2$, then $y = 3x + 2 \implies x = 3y + 2 \implies y = \frac{x - 2}{3}$, so $f^{-1}(x) = \frac{x - 2}{3}$.

Trigonometric Functions and Applications

Basic Trigonometric Functions

Trigonometric functions relate angles to side lengths in right triangles and are periodic.

• Sine, Cosine, Tangent: $\sin(\theta)$, $\cos(\theta)$, $\tan(\theta)$

• Periodicity: These functions repeat their values in regular intervals.

• Example: The graph of $y = \sin(x)$ oscillates between -1 and 1 with period $2\pi$.

Graphing and Identifying Functions

Graph Matching and Transformations

Identifying the correct graph for a given function involves understanding its basic shape and how transformations affect it.

• Linear Functions: Straight lines, slope-intercept form $y = mx + b$.

• Quadratic Functions: Parabolas, standard form $y = ax^2 + bx + c$.

• Piecewise and Absolute Value Functions: May have sharp corners or different behaviors in different intervals.

• Example: Matching $y = |x|$ to its V-shaped graph.

Tables: Summary of Function Properties

Property	Even Function	Odd Function	Neither
Algebraic Test	$f(-x) = f(x)$	$f(-x) = -f(x)$	Neither condition holds
Graph Symmetry	y-axis	Origin	None
Example	$x^2$	$x^3$	$x^2 + x$

Additional info:

• Some questions involve real-world modeling (e.g., fencing problems, sound waves), which are common applications in College Algebra.

• Trigonometric and piecewise functions are included as part of the standard College Algebra curriculum.