Inverse Functions and One-to-One Functions: Definitions, Properties, and Examples

Inverse Functions and One-to-One Functions

Introduction

This section explores the concept of inverse functions, how to determine if a function has an inverse, and the properties of one-to-one functions. These topics are foundational in Precalculus and are essential for understanding more advanced mathematical concepts, including calculus.

Definition of Inverse Functions

• Inverse Function: For a function f(x) with domain D and range R, its inverse function (if it exists) is a function f^{-1}(y) with domain R and range D such that:

$f^{-1}(f(x)) = x$ for all $x$ in $D$ $f(f^{-1}(y)) = y$ for all $y$ in $R$

• The inverse function essentially 'reverses' the effect of the original function.

• Not all functions have inverses. A function must be one-to-one to have an inverse.

One-to-One Functions

• Definition: A function is one-to-one (injective) if $f(x_1) \neq f(x_2)$ whenever $x_1 \neq x_2$.

• This means that each output value is produced by exactly one input value.

• Graphically, a function is one-to-one if any horizontal line intersects its graph at most once (Horizontal Line Test).

Motivation and Example

• Suppose a particle's velocity is modeled by $v(t) = 3t + 2$. To find the time when the velocity is $17$m/s, solve for$t$:

$17 = 3t + 2$ $t = \frac{17 - 2}{3} = 5$

• This process involves finding the inverse of the function $v(t)$.

Finding the Inverse of a Function

1. Replace $f(x)$ with $y$.

2. Interchange $x$ and $y$ in the equation.

3. Solve for $y$ in terms of $x$.

4. Replace $y$ with $f^{-1}(x)$.

Example: Find the inverse of $f(x) = 3x + 2$.

1. $y = 3x + 2$ 2. $x = 3y + 2$ 3. $y = \frac{x - 2}{3}$ 4. $f^{-1}(x) = \frac{x - 2}{3}$

Domain and Range of Inverse Functions

• The domain of $f$ becomes the range of $f^{-1}$, and the range of $f$ becomes the domain of $f^{-1}$.

• Example: For $f(x) = \sqrt[4]{x - 2}$, the domain is $x \geq 2$. The range is $y \geq 0$. The inverse function will have domain $y \geq 0$ and range $x \geq 2$.

Inverse Trigonometric Functions

• Domain of $\cos^{-1}(x)$: $-1 \leq x \leq 1$

• Example: $\cos^{-1}(\frac{1}{2}) = \frac{\pi}{3}$

• Composite Example: $\cos(\sin^{-1}(1))$ - $\sin^{-1}(1) = \frac{\pi}{2}$ - $\cos(\frac{\pi}{2}) = 0$

Common Mistakes

• When finding the inverse, always check the domain and range of both the original and inverse functions.

• Be careful with functions that are not one-to-one over their entire domain; restrict the domain if necessary to ensure invertibility.

Summary Table: Properties of Inverse Functions

Property	Original Function $f(x)$	Inverse Function $f^{-1}(x)$
Domain	$D$	$R$
Range	$R$	$D$
Composition	$f(f^{-1}(x)) = x$ for $x$ in $R$	$f^{-1}(f(x)) = x$ for $x$ in $D$
One-to-one requirement	Must be one-to-one	Always a function

Key Takeaways

• Only one-to-one functions have inverses that are also functions.

• To find an inverse, swap $x$ and $y$ and solve for $y$.

• Check the domain and range carefully, especially for trigonometric and root functions.