College Algebra: Functions, Graphs, and Linear Equations Study Notes

Basic Functions and Their Graphs

Rectangular Coordinates

In mathematics, any point in the plane can be represented by an ordered pair of numbers called rectangular coordinates (x, y). These coordinates are used to plot points on the Cartesian plane.

• Example: The point (4, -3) is 4 units to the right of the origin and 3 units down.

Relations and Functions

A relation is any set of ordered pairs. The domain is the set of all first components (x-values), and the range is the set of all second components (y-values).

• Example: For the relation {(1,2), (5,0), (7,-2)}, the domain is {1,5,7} and the range is {2,0,-2}.

Definition of a Function

A function is a relation in which each member of the domain corresponds to exactly one member of the range. If any x-value is paired with more than one y-value, the relation is not a function.

• Example: The relation {(1,2), (2,3), (3,4)} is a function, but {(1,2), (1,3)} is not.

Function Notation

Functions are often written as f(x), where x is the input value. For example, if $f(x) = x^2 + 4x + 10$, then $f(2) = 2^2 + 4(2) + 10 = 4 + 8 + 10 = 22$.

Identifying Functions from Equations

• To determine if an equation is a function, solve for y in terms of x and check if each x-value gives only one y-value.

• Example: $y = 3x^2$ is a function; $y^2 = x + 13$ is not a function because for some x-values, there are two possible y-values.

Intervals and Set Notation

Intervals describe sets of numbers, often used for domains and ranges.

Interval	Set Description	Graph
(a, b)	{x | a < x < b}	Open interval (no endpoints included)
[a, b]	{x | a ≤ x ≤ b}	Closed interval (endpoints included)
(a, b]	{x | a < x ≤ b}	Left open, right closed
[a, b)	{x | a ≤ x < b}	Left closed, right open
(-∞, ∞)	All real numbers	Entire real line

Functions and Their Graphs

Vertical Line Test

The vertical line test is used to determine if a graph represents a function. If any vertical line intersects the graph at more than one point, the graph does not represent a function.

• Example: A parabola opening upwards passes the vertical line test; a circle does not.

Evaluating Functions from Graphs

Given a graph of a function, you can find the value of f(x) for specific x-values by reading the corresponding y-value on the graph.

• Example: If the graph passes through (2, 1), then $f(2) = 1$.

Domain, Range, and Intercepts

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

• Range: The set of all possible y-values the function can take.

• x-intercept: The point(s) where the graph crosses the x-axis (y = 0).

• y-intercept: The point(s) where the graph crosses the y-axis (x = 0).

More on Functions and Their Graphs

Increasing and Decreasing Functions

A function is increasing on an interval if its graph rises as you move from left to right, and decreasing if it falls. A constant function remains flat.

• Relative Maximum: A point where the function changes from increasing to decreasing.

• Relative Minimum: A point where the function changes from decreasing to increasing.

Piecewise Functions

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

• Example: $f(x) = \begin{cases} x^2 & x < 0 \\ x+1 & x \geq 0 \end{cases}$

Difference Quotient

The difference quotient of a function f is given by:

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

This expression is fundamental in calculus for finding the slope of the secant line between two points on the graph of f.

Even and Odd Functions

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

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

• Neither: If neither condition is met, the function is neither even nor odd.

Linear Functions and Slope

The Slope of a Line

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

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

• Positive slope: Line rises left to right.

• Negative slope: Line falls left to right.

• Zero slope: Horizontal line.

• Undefined slope: Vertical line.

Equations of a Line

• Slope-intercept form: $y = mx + b$

• Point-slope form: $y - y_1 = m(x - x_1)$

• General form: $Ax + By + C = 0$

Parallel and Perpendicular Lines

• Parallel lines: Have the same slope ($m_1 = m_2$).

• Perpendicular lines: Slopes are negative reciprocals ($m_1 = -\frac{1}{m_2}$).

Average Rate of Change

The average rate of change of a function f from $x_1$ to $x_2$ is:

$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$

Combinations and Compositions of Functions

Operations on 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: $(\frac{f}{g})(x) = \frac{f(x)}{g(x)}$, $g(x) \neq 0$

Composite Functions

The composition of functions f and g is $(f \circ g)(x) = f(g(x))$.

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

Inverse Functions

Definition and Verification

The inverse of a function f, denoted $f^{-1}$, satisfies:

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

• To verify two functions are inverses, check both compositions yield x.

Finding the Inverse of a Function

1. Replace $f(x)$ with y.

2. Interchange x and y.

3. Solve for y.

4. Replace y with $f^{-1}(x)$.

• Example: $f(x) = 3x - 5$ 1. $y = 3x - 5$ 2. $x = 3y - 5$ 3. $y = \frac{x+5}{3}$ 4. $f^{-1}(x) = \frac{x+5}{3}$

Review Table: Interval Notation, Set Description, and Graphs

Interval	Set Description	Graph Description
(a, b)	{x | a < x < b}	Open interval (no endpoints included)
[a, b]	{x | a ≤ x ≤ b}	Closed interval (endpoints included)
(a, b]	{x | a < x ≤ b}	Left open, right closed
[a, b)	{x | a ≤ x < b}	Left closed, right open
(-∞, ∞)	All real numbers	Entire real line

Additional info: These notes cover the foundational concepts of College Algebra, including functions, their properties, graphing, and operations on functions. The content is suitable for exam preparation and as a reference for key algebraic principles.