Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Equations as Functions

Equations that Define Relations

Suppose we have an equation in two variables x and y. Let’s consider y = 4x + 3 and x2 - y2 = 1.

For any equation, the solution set is the set of ordered pairs (x, y) for which the equation holds true when the values are substituted for x and y. Since a set of ordered pairs is a relation, we can conclude that a solution set is a relation. In this way, equations represent relations.

A relation that can be represented by an equation can easily be visualized by graphing the equation. The graphs of y = 4x + 3 and x2 - y2 = 1 are shown.

When is a Relation a Function?

The set of first coordinates in a relation is called the domain of the relation. The set of second coordinates is called the range. Remember that a relation is a function if each element of the domain is paired with exactly one element in the range.

When we have a relation defined by an equation, we can graph the equation and then visually determine whether or not it is a function by using the vertical line test.

Key Idea

For a relation defined by an equation, the relation is a function if every vertical line intersects the graph of the equation in at most one point.

Consider the graph of y = 4x + 3. Using the vertical line test, we find that the equation does define a function.
As the vertical line moves to the right across the graph, it intersects only one point at a time.

Consider the graph of x2 - y2 = 1. Using the vertical line test, we find that the equation does not define a function.

As the vertical line moves to the right across the graph, it usually intersects the graph in two points. Sometimes it does not intersect the graph at all.

Recognizing Functions Algebraically

Another way to recognize when an equation defines a function is to solve for y in terms of x.

Key Idea Any equation which has y on one side and a single formula involving x on the other side defines a function.

The following equations are examples of functions.

Some equations are not functions. Consider x2 - y2 = 1. If x2 - y2 = 1 is solved for y, the result is The symbol ± indicates that there are two formulas on the right side of the equation for y. Hence, x2 - y2 = 1 does not represent a function.

If an equation is not written in the form y = (formula in x), the equation may still determine a function. Consider the following example.

In 4x - 2y = 10, y is not isolated on the left side. However, we can solve the equation for y.
4x - 2y

4x - 2y + 2x


= 10

= 10 - 4x

= - 4x + 10

Subtract 4x from each side.
Divide each side by -2.
y = 2x - 5  
This is now in the form y = (formula in x) and gives a function.

Key Idea If, in an equation, we can solve for y as a single formula in terms of x, then that equation represents a function.

Functions as Rules

Functions can also be thought of as rules that take an input value of x and produce an output value of y. We often give these rules names such as f, g, h, etc. We can then define a function by a formula in x.

The rule f(x) = x2 + 5 represents a function. We can think of it as a relation by taking the function to be the set of all ordered pairs of the form (x, f(x)) = (x, x2 + 5). It is also the solution set of y = x2 + 5.

Often we are given an equation in which y is not given as a formula in x, but we can solve for y in terms of x to get a rule defining the function.

In y + x = x2, we can solve for y by subtracting x from each side to get y = x2 - x.

This equation now gives a function. If we name the function f, then the function is defined by f(x) = x2 - x.

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