College Algebra: Functions and Graphs (Study Guide)

Functions and Graphs

Introduction to Functions and Their Graphs

This section introduces the foundational concepts of relations and functions, focusing on how to identify, represent, and analyze them using graphs and sets.

• Relation: A relation is a connection between two sets of values. It is often represented as a set of ordered pairs (x, y).

• Function: A function is a special type of relation in which each input (x-value) is paired with exactly one output (y-value).

• Vertical Line Test: A graph represents a function if and only if no vertical line intersects the graph at more than one point.

Example: The set {(1, 2), (2, 3), (3, 4)} is a function, but {(1, 2), (1, 3)} is not, since the input 1 is paired with two different outputs.

Domain and Range

The domain of a function is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

• To find the domain, look for all x-values for which the function is defined.

• To find the range, look for all y-values that the function can take.

• Interval notation and set-builder notation are commonly used to express domain and range.

Example: For $f(x) = \sqrt{x-2}$, the domain is $[2, \infty)$ because the expression under the square root must be non-negative.

Finding the Domain of an Equation

When given a function as an equation, determine the domain by identifying restrictions:

• Square Roots: The expression inside the square root must be non-negative.

• Denominators: The denominator cannot be zero.

Example: For $f(x) = \frac{1}{x-3}$, the domain is $(-\infty, 3) \cup (3, \infty)$.

Graphs of Common Functions

Several basic functions and their graphs are fundamental in algebra:

Function	Equation	Domain	Range
Constant	$f(x) = c$	$(-\infty, \infty)$	$\{c\}$
Identity	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$
Square	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Cube	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$
Cube Root	$f(x) = \sqrt[3]{x}$	$(-\infty, \infty)$	$(-\infty, \infty)$

Transformations of Functions

Transformations change the position and/or shape of a function's graph. The main types are reflections, shifts, and stretches/shrinks.

• Reflection: Flips the graph over a specified axis.

• Shift: Moves the graph horizontally or vertically.

• Stretch/Shrink: Changes the graph's size vertically or horizontally.

General Forms:

• Vertical shift: $f(x) + k$

• Horizontal shift: $f(x - h)$

• Reflection over x-axis: $-f(x)$

• Reflection over y-axis: $f(-x)$

• Vertical stretch/shrink: $a f(x)$

• Horizontal stretch/shrink: $f(bx)$

Example: $g(x) = -f(x-2) + 3$ is $f(x)$ reflected over the x-axis, shifted right 2 units, and up 3 units.

Function Operations

Functions can be added, subtracted, multiplied, or divided to create new functions.

• Addition: $(f+g)(x) = f(x) + g(x)$

• Subtraction: $(f-g)(x) = f(x) - g(x)$

• Multiplication: $(f \cdot g)(x) = f(x) \cdot g(x)$

• Division: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Domain of Combined Functions: The domain is the intersection of the domains of the original functions, with additional restrictions for division (denominator cannot be zero).

Function Composition

Function composition involves applying one function to the result of another: $(f \circ g)(x) = f(g(x))$.

• To evaluate, substitute $g(x)$ into $f(x)$.

• The domain of $f \circ g$ consists of all $x$ in the domain of $g$ such that $g(x)$ is in the domain of $f$.

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

Decomposing Functions

Decomposing a function means expressing it as a composition of two or more simpler functions.

• There are often multiple correct ways to decompose a function.

Example: $h(x) = \sqrt{2x+1}$ can be written as $h(x) = f(g(x))$ where $g(x) = 2x+1$ and $f(x) = \sqrt{x}$.

Circles in Standard Form

The equation of a circle in standard form is $(x-h)^2 + (y-k)^2 = r^2$, where $(h, k)$ is the center and $r$ is the radius.

• To graph a circle, identify the center and radius from the equation.

• If the equation is not in standard form, complete the square to rewrite it.

Example: $(x-2)^2 + (y+3)^2 = 9$ is a circle centered at (2, -3) with radius 3.

General Form to Standard Form for Circles

To convert a circle's equation from general form $x^2 + y^2 + Dx + Ey + F = 0$ to standard form, complete the square for both $x$ and $y$ terms.

• Group $x$ and $y$ terms, complete the square, and rewrite in standard form.

Example: $x^2 + y^2 - 4x + 6y + 9 = 0$ becomes $(x-2)^2 + (y+3)^2 = 4$ after completing the square.

Summary Table: Common Function Graphs

Type	Equation	Domain	Range
Constant	$f(x) = c$	$(-\infty, \infty)$	$\{c\}$
Identity	$f(x) = x$	$(-\infty, \infty)$	$(-\infty, \infty)$
Square	$f(x) = x^2$	$(-\infty, \infty)$	$[0, \infty)$
Cube	$f(x) = x^3$	$(-\infty, \infty)$	$(-\infty, \infty)$
Square Root	$f(x) = \sqrt{x}$	$[0, \infty)$	$[0, \infty)$
Cube Root	$f(x) = \sqrt[3]{x}$	$(-\infty, \infty)$	$(-\infty, \infty)$

Additional info: Practice problems and worked examples are included throughout the notes to reinforce understanding and application of these concepts.