Inverse Functions

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 $x_{1}{\text{ and }}x_{2}$ are two numbers in the domain of f such that $f(x_{1})=f(x_{2})$ then we must have that $x_{1}=x_{2}$

Examples: $x,x^{3},3x+2,$

Non-examples: $x^{2},\left|x\right|$

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 $f^{-1}$.

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

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 $f^{-1}(x)$ and check that $f^{-1}(f(x))=x{\text{ and }}f(f^{-1}(x))=x$

Example: Find the inverse of $f(x)={\frac {1}{x+3}}$

We start by swapping the x and y to get $x={\frac {1}{y+3}}$

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

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