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==Introduction== One-to-one functions, or invertible functions, are functions that have inverses. One-to-one functions have a nice test t..."

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==Introduction==

One-to-one functions, or invertible functions, are functions that have inverses. One-to-one functions have a nice test to tell from the graph if a function is one-to-one. The definition of a one-to-one function is
given through the equation of the function. The conceptual idea for a function being one-to-one is that for every number in the range there is one number in the domain that the function maps to the point in the range.

==Definition==

A function is one-to-one if two distinct values in the domain get mapped to the same value in the range. More concretely, if f is one-to-one and <math>x_1 \text{ and }x_2</math> are two numbers in the domain of f such that <math>f(x_1) = f(x_2)</math>
then we must have that <math>x_1 = x_2</math>

Examples: <math> x, x^3, 3x + 2, </math>

Non-examples: <math> x^2, \left| x \right|</math>

==Horizontal Line Test==

It was mentioned earlier that there is a way to tell if a function is one-to-one from its graph. This method is called the horizontal line test.
It is the same as the vertical line test, except we use a horizontal line. So a function is one-to-one if every horizontal line crosses the graph
at most once.

==Inverse Functions==

If a function is one-to-one, then to each y in the range of f there is a unique x in the domain that maps on top of it. So we can define a
function from the range of f back to the domain. This new function is called the inverse function, and is denoted <math> f^{-1}</math>.

Properties:
Let f be a one-to-one function, and <math>f^{-1}</math> be its inverse.
a) Domain of f = Range of <math>f^{-1}</math>
b) Range of f = Domain of <math>f^{-1}</math>
c) <math> f^{-1}(f(x)) = x</math> where x is in the domain of f^{-1}
d) <math> f(f^{-1}(x)) = x </math> where x is in the domain of <math> f^{-1}(x)</math>

Since we have both a graphical and algebraic method to determine if a function is one-to-one, we also have geometric and algebraic methods to find the inverse.

The graphical method is to reflect the graph of f across the line y = x. Although easy to explain, the process is harder to visualize.
The algebraic method is the opposite, harder to explain but easier to execute.

The algebraic method can be explained in 3 steps:

Step 1) In the equation y = f(x), interchange all instances of x with the variable y, and vice versa.

Step 2) If possible, solve for y.

Step 3) Replace y with <math>f^{-1}(x)</math> and check that <math>f^{-1}(f(x)) = x \text{ and }f(f^{-1}(x)) = x</math>

Example:
Find the inverse of <math>f(x) = \frac{1}{x + 3}</math>

We start by swapping the x and y to get <math>x = \frac{1}{y + 3}</math>

Now we can solve for y and swap y for <math>f^{-1}(x)</math>. Doing so we will find that <math>f^{-1}(x) = \frac{1-3x}{x}</math>

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Inverse Functions - Revision history

MathAdmin: Created page with "__TOC__ ==Introduction== One-to-one functions, or invertible functions, are functions that have inverses. One-to-one functions have a nice test t..."

MathAdmin: Created page with "
TOC
==Introduction== One-to-one functions, or invertible functions, are functions that have inverses. One-to-one functions have a nice test t..."