Algebra Tutorials!
Saturday 15th of June
Calculations with Negative Numbers
Solving Linear Equations
Systems of Linear Equations
Solving Linear Equations Graphically
Algebra Expressions
Evaluating Expressions and Solving Equations
Fraction rules
Factoring Quadratic Trinomials
Multiplying and Dividing Fractions
Dividing Decimals by Whole Numbers
Adding and Subtracting Radicals
Subtracting Fractions
Factoring Polynomials by Grouping
Slopes of Perpendicular Lines
Linear Equations
Roots - Radicals 1
Graph of a Line
Sum of the Roots of a Quadratic
Writing Linear Equations Using Slope and Point
Factoring Trinomials with Leading Coefficient 1
Writing Linear Equations Using Slope and Point
Simplifying Expressions with Negative Exponents
Solving Equations 3
Solving Quadratic Equations
Parent and Family Graphs
Collecting Like Terms
nth Roots
Power of a Quotient Property of Exponents
Adding and Subtracting Fractions
Solving Linear Systems of Equations by Elimination
The Quadratic Formula
Fractions and Mixed Numbers
Solving Rational Equations
Multiplying Special Binomials
Rounding Numbers
Factoring by Grouping
Polar Form of a Complex Number
Solving Quadratic Equations
Simplifying Complex Fractions
Common Logs
Operations on Signed Numbers
Multiplying Fractions in General
Dividing Polynomials
Higher Degrees and Variable Exponents
Solving Quadratic Inequalities with a Sign Graph
Writing a Rational Expression in Lowest Terms
Solving Quadratic Inequalities with a Sign Graph
Solving Linear Equations
The Square of a Binomial
Properties of Negative Exponents
Inverse Functions
Rotating an Ellipse
Multiplying Numbers
Linear Equations
Solving Equations with One Log Term
Combining Operations
The Ellipse
Straight Lines
Graphing Inequalities in Two Variables
Solving Trigonometric Equations
Adding and Subtracting Fractions
Simple Trinomials as Products of Binomials
Ratios and Proportions
Solving Equations
Multiplying and Dividing Fractions 2
Rational Numbers
Difference of Two Squares
Factoring Polynomials by Grouping
Solving Equations That Contain Rational Expressions
Solving Quadratic Equations
Dividing and Subtracting Rational Expressions
Square Roots and Real Numbers
Order of Operations
Solving Nonlinear Equations by Substitution
The Distance and Midpoint Formulas
Linear Equations
Graphing Using x- and y- Intercepts
Properties of Exponents
Solving Quadratic Equations
Solving One-Step Equations Using Algebra
Relatively Prime Numbers
Solving a Quadratic Inequality with Two Solutions
Operations on Radicals
Factoring a Difference of Two Squares
Straight Lines
Solving Quadratic Equations by Factoring
Graphing Logarithmic Functions
Simplifying Expressions Involving Variables
Adding Integers
Factoring Completely General Quadratic Trinomials
Using Patterns to Multiply Two Binomials
Adding and Subtracting Rational Expressions With Unlike Denominators
Rational Exponents
Horizontal and Vertical Lines
Try the Free Math Solver or Scroll down to Tutorials!












Please use this form if you would like
to have this math solver on your website,
free of charge.

Inverse Functions

Sometimes one function can "undo" what another function does. For example, consider f(x) = 5x over the domain {1, 2, 3}

1 → 5

2 → 10

3 → 15

Now consider over the domain {5, 10, 15}

5 1

10 → 2

15 → 3

Note: f takes 2 to 10, and g takes 10 back to 2.

This is an example of a pair of functions that are inverses of each other.

Before we officially define inverses, we need another definition...

One-to-One Function:

A one-to-one function has exactly one output for each input and exactly one input for each output. This means that in addition to the fact that no two points can have the same first coordinate, in order to be a one-to-one function, it must also be true that no two points can have the same second coordinate. Graphically, we can use the Horizontal Line Test to determine if a function is one-to-one.


Horizontal Line Test: A function is one-to-one if no horizontal line intersects its graph in more than one point.


Now, back to inverses. It is true that every one-to-one function has an inverse. Now we are ready to  define inverse functions.

DEFINITION: Functions f and g for which f(g(x)) = x and g(f(x)) = x for all x in the domains of f and g are called inverse functions. We denote the inverse of f by f-1, read "f inverse".

Our next question is: "How do we go about finding the inverse of a one-to-one function?" There is a four step process for finding a function inverse.


Finding the Inverse of a Function

To find the inverse of a one-to-one function defined by the equation y = f(x):

1. Rewrite the equation, replacing f(x) with y.

2. Interchange x and y.

3. Solve the resulting equation for y. (If this equation cannot be solved uniquely for y, the original function has no inverse.)

4. Replace y with f-1(x).


Verifying That Two Functions Are Inverses

To verify that two functions are inverses, you must show that both of the following are true:

f(f-1(x)) = x and f-1(f(x)) = x

The domain of a function will become the range of its inverse and the range of a function will become the domain of its inverse.

Copyrights © 2005-2024