Foundations of Relations, Functions, and Equations in College Algebra

Things To Know: Sets, Relations, and Functions

Writing Sets Using Set-Builder Notation and Interval Notation

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

• Set-Builder Notation: Describes a set by stating the properties its members must satisfy. Example: $\{x \mid x > 0\}$ means all x such that x is greater than 0.

• Interval Notation: Uses intervals to describe sets of numbers. Example: $(0, \infty)$ means all real numbers greater than 0.

Application: Use these notations to express solution sets for equations and inequalities.

Solving Equations and Inequalities

Solving Rational Equations That Lead to Linear Equations

Rational equations involve fractions with variables in the denominator. To solve, clear denominators and solve the resulting equation, checking for extraneous solutions.

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

Steps:

1. Multiply both sides by the common denominator $(x-2)$ to eliminate fractions.

2. Solve the resulting linear equation.

3. Check for restricted values (values that make the denominator zero).

4. Exclude any extraneous solutions.

In the example above: $x = 2$ is a restricted value, so there is no solution.

Solving Quadratic Equations by Factoring and the Zero Product Property

Quadratic equations can often be solved by factoring and applying the zero product property.

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$.

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

Steps:

1. Factor the quadratic: $6x^2 - 17x + 12 = (2x-3)(3x-4) = 0$

2. Set each factor to zero and solve:

   • $2x-3=0 \implies x=\frac{3}{2}$

   • $3x-4=0 \implies x=\frac{4}{3}$

Solution: $x = \frac{3}{2}, \frac{4}{3}$

Solving Linear Inequalities

Linear inequalities are solved similarly to linear equations, but the direction of the inequality reverses when multiplying or dividing by a negative number.

• Example: Solve $-9x - 3 \geq 7 - 4x$

Steps:

1. Isolate x on one side.

2. Solve for x, remembering to reverse the inequality if multiplying/dividing by a negative.

Solution: $x \leq -2$

Solving Polynomial Inequalities

Polynomial inequalities involve finding intervals where a polynomial is positive or negative.

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

Steps:

1. Factor the polynomial: $x(x-1)(x-2) \geq 0$

2. Find the zeros: $x=0, 1, 2$

3. Test intervals between zeros to determine where the expression is nonnegative.

4. Write the solution in interval notation.

Solution: $[0,1] \cup [2, \infty)$

Graphing Equations by Plotting Points

Plotting Points to Graph Quadratic Equations

To graph a quadratic equation, calculate y-values for selected x-values and plot the resulting points.

• Example: Graph $y = x^2 - 4x + 4$ for $x = -1, 0, 1, 2, 3, 4$

x	y	Point (x, y)
-1	9	(-1, 9)
0	4	(0, 4)
1	1	(1, 1)
2	0	(2, 0)
3	1	(3, 1)
4	4	(4, 4)

Application: Plot these points and connect them to form the parabola.

Relations and Functions

Definition of a Relation

A relation is a correspondence between two sets, A and B, such that each element of set A is paired with one or more elements of set B. Set A is called the domain and set B is called the range of the relation.

• Example: If A = {a, b, c, d} and B = {e, f, g}, a relation could be {(a, e), (b, f), (c, f), (d, g)}

• Domain: {a, b, c, d}

• Range: {e, f, g}

Definition of a Function

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

• Example: If A = {a, b, c, d} and B = {e, f, g, h}, a function could be {(a, e), (b, f), (c, g), (d, h)}

• Each input (domain) has only one output (range).

Determining Whether a Relation is a Function

To determine if a relation is a function, check that no element in the domain is paired with more than one element in the range.

• Example: The set {(0, 7), (1, 7), (2, 7), (6, 7)} is a function because each domain value is paired with only one range value.

• The set {(4, -1), (0, 1), (1, 5), (0, 3)} is not a function because 0 in the domain is paired with both 1 and 3.

Domain and Range from Ordered Pairs and Graphs

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

• Example: For the set {(3, 7), (-3, 2), (-4, 5), (1, 4), (3, -4)}:

  • Domain: {3, -3, -4, 1}

  • Range: {7, 2, 5, 4, -4}

  • Not a function because 3 in the domain is paired with both 7 and -4.

Determining Whether Equations Represent Functions

Testing Equations for Functionality

An equation represents y as a function of x if for every x-value, there is exactly one corresponding y-value.

• Example: $3x - 2y = -12$

• Solve for y: $y = \frac{3x + 12}{2}$

• For every x, there is exactly one y, so this is a function.

• Example: $y^2 = 4x - 16$

• Solve for y: $y = \pm \sqrt{4x - 16}$

• For some x-values, there are two y-values, so this is not a function.

Summary Table: Relation vs. Function

Aspect	Relation	Function
Definition	Pairs elements of two sets	Each input has exactly one output
Domain	All first elements	All first elements (no repeats with different outputs)
Range	All second elements	All second elements paired with domain
Example	{(a, e), (a, f)}	{(a, e), (b, f)}

Key Formulas and Properties

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

• Zero Product Property: If $ab = 0$, then $a = 0$ or $b = 0$

• Interval Notation: $(a, b)$, $[a, b]$, $(-\infty, a)$, $(b, \infty)$, etc.

Additional info:

• Set-builder and interval notation are foundational for expressing domains and solution sets in algebra.

• Understanding the distinction between relations and functions is critical for later topics such as function composition and inverse functions.