College Algebra Study Notes: Equations, Inequalities, and Functions

Things To Know: Sets, Equations, Inequalities, and Functions

Writing Sets Using Set-Builder Notation and Interval Notation

In algebra, sets of numbers are often described using set-builder notation and interval notation. These notations help specify the elements of a set, especially when describing solution sets for equations and inequalities.

• Set-builder notation: Describes a set by stating the properties that its members must satisfy. Example: {x | x > 2} means "the set of all x such that x is greater than 2."

• Interval notation: Uses intervals to describe subsets of real numbers. Example: (2, ∞) represents all real numbers greater than 2.

Additional info: Review of set-builder and interval notation is foundational for expressing solution sets in algebra.

Solving Equations and Inequalities

Solving Rational Equations that Lead to Linear Equations

Rational equations involve fractions with polynomials in the numerator and denominator. To solve, clear denominators and solve the resulting linear equation.

• Example: Solve $\frac{2x}{x+2} + 3 = \frac{4}{x+2}$

• Multiply both sides by (x+2) to clear denominators:

• $2x + 3(x+2) = 4$

• Expand and solve for x:

• $2x + 3x + 6 = 4$

• $5x + 6 = 4$

• $5x = -2$

• $x = -\frac{2}{5}$

Solving Quadratic Equations by Factoring and the Zero Product Property

Quadratic equations can often be solved by factoring and applying the zero product property, which states that if $ab = 0$, then $a = 0$ or $b = 0$.

• Example: Solve $6x^2 - 17x + 12 = 0$

• Factor: $(2x-3)(3x-4) = 0$

• Set each factor to zero:

• $2x-3=0 \implies x=\frac{3}{2}$

• $3x-4=0 \implies x=\frac{4}{3}$

Solving Linear Inequalities

To solve inequalities, isolate the variable using algebraic operations, remembering to reverse the inequality sign when multiplying or dividing by a negative number.

• Example: Solve $-5x - 3 \geq 7 - 4x$

• Add $4x$ to both sides: $-5x + 4x - 3 \geq 7$

• $-x - 3 \geq 7$

• Add 3: $-x \geq 10$

• Divide by -1 (reverse sign): $x \leq -10$

Solving Polynomial Inequalities

Polynomial inequalities are solved by finding the zeros of the polynomial, plotting them on a number line, and testing intervals to determine where the inequality holds.

• Example: Solve $x^2 - 3x + 2 \geq 0$

• Factor: $(x-1)(x-2) \geq 0$

• Zeros at $x=1$ and $x=2$

• Test intervals: $(-\infty,1)$, $(1,2)$, $(2,\infty)$

• Solution: $(-\infty,1] \cup [2,\infty)$

Graphing Equations and Functions

Graphing Equations by Plotting Points

To graph an equation, create a table of values by substituting values for x and finding the corresponding y-values, then plot the points and connect them smoothly.

• Example: Graph $y = x^2 - 4x + 4$

• Table of values:

x	y
0	4
1	1
2	0
3	1
4	4

• Plot these points and draw a parabola.

Functions and Relations

Definition of a Relation

A relation is a correspondence between two sets, where each element of the first set (domain) is paired with one or more elements of the second set (range).

• Domain: The set of all possible input values (x-values).

• Range: The set of all possible output values (y-values).

• Example: If A = {a, b, c} and B = {e, f, g}, a relation could pair a with e, b with f, and c with g.

Definition of a Function

A function is a special type of relation in which each element of the domain is paired with exactly one element of the range.

• Key property: No input (x-value) is paired with more than one output (y-value).

Determining Functions from Relations

To determine if a relation is a function, check that each input has only one output.

• Example: Relation: { (Phoebe, 5'5"), (Amy, 3'1"), (Catie, 3'1") } is a function because each name is paired with only one height.

• If any input is paired with more than one output, the relation is not a function.

Domain and Range from Graphs and Ordered Pairs

Given a set of ordered pairs or a graph, the domain is the set of all first elements (x-values), and the range is the set of all second elements (y-values).

• Example: For the set {(-4,5), (-2,2), (0,6), (1,-2), (2,4)}, the domain is {-4, -2, 0, 1, 2} and the range is {5, 2, 6, -2, 4}.

Determining Whether Equations Represent Functions

To determine if an equation represents y as a function of x, solve for y and check if each x-value yields only one y-value.

• Example: $3x - 2y = -12$

• Solve for y: $y = \frac{3x+12}{2}$

• For each x, there is only one y, so it is a function.

Function Notation and Evaluating Functions

Function notation uses symbols like f(x) to denote the output of a function f for input x. To evaluate, substitute the given value for x.

• Example: If $f(x) = \frac{3}{2}x + 6$, then $f(2) = \frac{3}{2}(2) + 6 = 3 + 6 = 9$.

Independent and Dependent Variables

• Independent variable: The variable whose value determines the value of other variables (usually x).

• Dependent variable: The variable whose value depends on the independent variable (usually y or f(x)).

Vertical Line Test

The vertical line test is a graphical method to determine if a curve is the graph of a function. If any vertical line crosses the graph more than once, the graph does not represent a function.

• Rule: No vertical line should intersect the graph at more than one point.

• Example: The graph of $y = x^2$ passes the vertical line test and is a function; a circle does not.

Summary Table: Key Concepts

Concept	Definition	Example
Set-builder notation	Describes a set by a property	{x | x > 0}
Interval notation	Describes a set as an interval	(0, ∞)
Function	Each input has one output	f(x) = x^2
Relation	Any pairing of inputs and outputs	{(1,2), (2,3)}
Vertical line test	Test for function from graph	Parabola passes, circle fails

Additional info: These foundational concepts are essential for success in College Algebra and are frequently tested on exams.