Problem 15 - Essential Calculus Early Transcendentals

Question

Assume that \(f\) is a one-to-one function. (a) If \(f(6)=17,\) what is \(f^{-1}(17) ?\) (b) If \(f^{-1}(3)=2,\) what is \(f(2) ?\)

Step-by-Step Solution

Answer

(a) 6 (b) 3

Step 1: Understanding the Definition of Inverse Functions

A function \(f\) is called one-to-one if it passes the horizontal line test, meaning every \(y\) value has exactly one corresponding \(x\) value. The inverse of \(f\), denoted as \(f^{-1}\), essentially reverses this relationship.

Step 2: Solving Part (a)

We know that \(f(6)=17\). The property of inverse functions tells us that if \(f(a)=b\), then \(f^{-1}(b)=a\). Therefore, since \(f(6)=17\), it follows that \(f^{-1}(17)=6\).

Step 3: Solving Part (b)

The problem states \(f^{-1}(3)=2\). Using the definition of inverse functions, this means \(f(2)=3\), since if \(f^{-1}(b)=a\), then \(f(a)=b\).

Key Concepts

One-to-One Function

To understand inverse functions, we start with the concept of a "one-to-one function." In simple terms, a one-to-one function ensures that every output (or second element in a pair) is paired uniquely with one input (or first element in a pair).
Imagine matching key to keyhole, where each key fits exactly one specific keyhole. In the context of functions:

• Each value of the function's output corresponds to only one input.
• This uniqueness allows us to "reverse" the function with an inverse function.

If a function is not one-to-one, some outputs might be linked to more than one input, making it impossible to trace back uniquely. That's why one-to-one functions are a prerequisite for inverse functions.

Horizontal Line Test

To determine if a function is one-to-one, we can use a simple visual tool called the "Horizontal Line Test." This test allows us to see if a function qualifies for having an inverse.
The procedure is straightforward:

• Imagine drawing horizontal lines across the graph of the function.
• If any horizontal line intersects the graph more than once, the function is not one-to-one.
• If each horizontal line intersects the graph at most once, then the function is one-to-one.

This test is essential because it helps us to visually confirm the possibility of defining an inverse function. If a function passes this test, it confirms the "key and keyhole" scenario we described earlier.

Function Properties

Understanding function properties is key in working with inverses. Besides being one-to-one, knowing other properties can help to deal with inverse functions gracefully.
Here are some properties to consider:

• Domain and Range: For a function and its inverse, the domain of the function becomes the range of the inverse and vice versa.
• Symmetry: Graphically, a function and its inverse are mirror images of each other across the line \( y = x \).

These properties provide foundational understanding in working with inverse functions. They establish the parameters which must be considered to perform operations correctly.

Inverse Relationship

An inverse relationship in mathematical functions is a fundamental concept where the operation can be "reversed." Inverse functions \( f \, \text{and} \, f^{-1} \) are pairs that satisfy this relationship.
Here's what to keep in mind:

• If \( f(a) = b \), then \( f^{-1}(b) = a \).
• This relationship allows us to determine unknowns. For example, if you know \( f(2) = 3 \), then \( f^{-1}(3) = 2 \).
• The notation \( f^{-1} \) does not mean \( 1/f \); it specifically represents the unique function that "undoes" \( f \).

Understanding inverse relationships is about maneuvering back and forth between inputs and outputs, akin to finding your way out and back in through a maze. Keep these inverse concepts in mind when solving related problems.