Systems of Equations
Adding and Subtracting Rational Expressions with Different Denominators
Graphing Linear Equations
Raising an Exponential Expression to a Power
Horizontal Line Test
Quadratic Equations
Mixed Numbers and Improper Fractions
Solving Quadratic Equations by Completing the Square
Solving Exponential Equations
Adding and Subtracting Polynomials
Factorizing simple expressions
Identifying Prime and Composite Numbers
Solving Linear Systems of Equations by Graphing
Complex Conjugates
Graphing Compound Inequalities
Simplified Form of a Square Root
Solving Quadratic Equations Using the Square Root Property
Multiplication Property of Radicals
Determining if a Function has an Inverse
Scientific Notation
Degree of a Polynomial
Factoring Polynomials by Grouping
Solving Linear Systems of Equations
Exponential Functions
Factoring Trinomials by Grouping
The Slope of a Line
Simplifying Complex Fractions That Contain Addition or Subtraction
Solving Absolute Value Equations
Solving Right Triangles
Solving Rational Inequalities with a Sign Graph
Domain and Range of a Function
Multiplying Polynomials
Slope of a Line
Multiplying Rational Expressions
Percent of Change
Equations Involving Fractions or Decimals
Simplifying Expressions Containing only Monomials
Solving Inequalities
Quadratic Equations with Imaginary Solutions
Reducing Fractions to Lowest Terms
Prime and Composite Numbers
Dividing with Exponents
Dividing Rational Expressions
Equivalent Fractions
Graphing Quadratic Functions
Linear Equations and Inequalities in One Variable
Notes on the Difference of 2 Squares
Solving Absolute Value Inequalities
Solving Quadratic Equations
Factoring Polynomials Completely
Using Slopes to Graph Lines
Fractions, Decimals and Percents
Solving Systems of Equations by Substitution
Quotient Rule for Radicals
Prime Polynomials
Solving Nonlinear Equations by Substitution
Simplifying Radical Expressions Containing One Term
Factoring a Sum or Difference of Two Cubes
Finding the Least Common Denominator of Rational Expressions
Multiplying Rational Expressions
Expansion of a Product of Binomials
Solving Equations
Exponential Growth
Factoring by Grouping
Solving One-Step Equations Using Models
Solving Quadratic Equations by Factoring
Adding and Subtracting Polynomials
Rationalizing the Denominator
Rounding Off
The Distributive Property
What is a Quadratic Equation
Laws of Exponents and Multiplying Monomials
The Slope of a Line
Factoring Trinomials by Grouping
Multiplying and Dividing Rational Expressions
Solving Linear Inequalities
Multiplication Property of Exponents
Multiplying and Dividing Fractions 3
Dividing Monomials
Multiplying Polynomials
Adding and Subtracting Functions
Dividing Polynomials
Absolute Value and Distance
Multiplication and Division with Mixed Numbers
Factoring a Polynomial by Finding the GCF
Adding and Subtracting Polynomials
The Rectangular Coordinate System
Polar Form of a Complex Number
Exponents and Order of Operations
Graphing Horizontal and Vertical Lines
Invariants Under Rotation
The Addition Method
Solving Linear Inequalities in One Variable
The Pythagorean Theorem
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Horizontal Line Test

A function where each y-value (output) corresponds to exactly one x-value (input) is called a one-to-one function. A one-to-one function has an inverse function.

To determine if a function is one-to-one, we can use the horizontal line test.


Another way to think about a function that is one-to-one is the following: Two different input values always result in 2 different output values.


Procedure — To Determine if a Graph Represents a One-To-One Function (Horizontal Line Test)

If you can draw a horizontal line anywhere through the graph of a function and it intersects the graph at most once, then the graph represents a one-to-one function.


Example 1

Given the graph of the function: f(x) = -2x + 4

a. Is f(x) a one-to-one function?

b. Does f(x) have an inverse?


a. To determine if the function is one-to-one, use the horizontal line test. Any horizontal line intersects the graph at most once. Thus, each output corresponds to at most one input.

Therefore, f(x) = -2x + 4 is a one-to-one function.

b. Since the function is one-to-one, f(x) has an inverse.


Any linear function is one-to-one unless it has the form f(x) =  c, where c is a constant.

The graph of f(x) = c is a horizontal line and so it does not pass the horizontal line test.


Example 2

Given the graph of the function: f(x) = -1(x - 3)2 + 4

a. Is f(x) a one-to-one function?

b. Does f(x) have an inverse?


a. To determine if the function is one-to-one, use the horizontal line test. Since a horizontal line may intersect the graph more than once, some outputs correspond to more than one input.

For example, the y-value 3 corresponds to two x-values (2 and 4). Therefore, this function is not one to one.

b. Since the function is not one-to-one, it does not have an inverse that is a function.

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