Study Guide: Functions, Graphs, and Piecewise Functions in College Algebra

Functions and Their Graphs

Even and Odd Functions

Understanding the symmetry of functions is essential in algebra. Functions can be classified as even, odd, or neither based on their symmetry properties.

• Even Function: A function f is even if f(-x) = f(x) for all x in the domain. The graph is symmetric with respect to the y-axis.

• Odd Function: A function f is odd if f(-x) = -f(x) for all x in the domain. The graph is symmetric with respect to the origin.

• Neither: If a function does not satisfy either condition, it is neither even nor odd.

Example: The function f(x) = x^2 is even, while f(x) = x^3 is odd.

Symmetry of Graphs

To determine symmetry:

• y-axis: Replace x with -x and check if the equation is unchanged.

• x-axis: Replace y with -y and check if the equation is unchanged.

• Origin: Replace (x, y) with (-x, -y) and check if the equation is unchanged.

Intervals of Increase, Decrease, and Constancy

Definitions

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

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

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

Example: For the function f(x) = x^2, it is decreasing on $(-\infty, 0)$ and increasing on $(0, \infty)$.

Relative Maximum and Minimum

Definitions

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

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

To find these points, look for where the function changes from increasing to decreasing (maximum) or decreasing to increasing (minimum).

Example: For f(x) = -x^2 + 4x - 1, the vertex is a relative maximum.

Piecewise Functions

Definition and Evaluation

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

• To evaluate, determine which interval the input belongs to and use the corresponding formula.

Example: If $f(x) = \begin{cases} x^2 & x < 0 \\ x + 1 & x \geq 0 \end{cases}$, then $f(-2) = 4$ and $f(1) = 2$.

Graphing Piecewise Functions

• Graph each piece on its specified interval.

• Pay attention to open and closed circles to indicate whether endpoints are included.

Quadratic Functions and the Quadratic Formula

Quadratic Formula

The quadratic formula solves equations of the form $ax^2 + bx + c = 0$:

$x = \frac{ -b \pm \sqrt{b^2 - 4ac} }{2a}$

Example: Solve $2x^2 - 5x + 3 = 0$ using the formula.

Difference Quotient

Definition

The difference quotient of a function f is:

$\frac{f(x+h) - f(x)}{h}$

This is used to compute the average rate of change and is foundational for calculus.

Tables: Symmetry and Function Classification

Equation	Even/Odd/Neither	Symmetry
$y = x^2$	Even	y-axis
$y = x^3$	Odd	Origin
$y = x^2 + 1$	Even	y-axis
$y = x^3 + x$	Odd	Origin
$y = x^2 + x$	Neither	None

Domain and Range

Definitions

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

• Range: The set of all possible output values (y-values) the function can produce.

Example: For $f(x) = x^2$, the domain is $(-\infty, \infty)$ and the range is $[0, \infty)$.

Applications: Real-World Piecewise Functions

Piecewise functions can model real-world scenarios, such as cell phone billing plans where the cost changes after a certain number of minutes.

Example: A plan charges $20 for up to 300 minutes, then $0.10 for each additional minute. The cost function is:

$C(m) = \begin{cases} 20 & 0 \leq m \leq 300 \\ 20 + 0.10(m - 300) & m > 300 \end{cases}$

Summary Table: Intervals of Increase/Decrease/Constancy

Interval	Behavior
$(-\infty, a)$	Increasing/Decreasing/Constant (depends on function)
$(a, b)$	Increasing/Decreasing/Constant (depends on function)
$(b, \infty)$	Increasing/Decreasing/Constant (depends on function)

Additional info: Some explanations and examples were expanded for clarity and completeness, including the use of tables and real-world applications.