Problem 22 - Algebra and Trigonometry

Question

Assume that \(f\) is a one-to-one function. $$ \begin{array}{l}{\text { (a) If } f(5)=18, \text { find } f^{-1}(18)} \\\ {\text { (b) If } f^{-1}(4)=2, \text { find } f(2)}\end{array} $$

Step-by-Step Solution

Answer

(a) \(f^{-1}(18) = 5\); (b) \(f(2) = 4\).

Step 1: Understanding Inverse Functions

The inverse function, denoted as \(f^{-1}\), reverses the mapping of the original function \(f\). If \(f(a) = b\), then \(f^{-1}(b) = a\). This property will help us solve both parts of the exercise.

Step 2: Solving Part (a)

We know that \(f(5) = 18\). By the definition of the inverse function, this means \(f^{-1}(18) = 5\). Hence, when we input 18 into \(f^{-1}\), we must obtain 5 because \(f\) and \(f^{-1}\) are inverse operations.

Step 3: Solving Part (b)

We are given \(f^{-1}(4) = 2\). According to the property of inverse functions, this implies that \(f(2) = 4\). Thus, to find \(f(2)\), we directly refer to the given inverse function information.

Key Concepts

Understanding One-to-One Functions

A one-to-one function (or injective function) is one of the foundational ideas in understanding inverse functions. This type of function ensures that each element of the function's domain maps to a unique element in the codomain. In other words:

• There are no repeated outputs for different inputs.
• If \( f(a) = f(b) \), then \( a = b \).
• Every input matches exactly one output, and every output comes from exactly one input.

Why is this important? Because only one-to-one functions have inverses that are also functions. If a function is not one-to-one, some outputs might map back to multiple inputs, making it impossible to reverse the process without losing information. This makes one-to-one functions particularly interesting for problems involving inverse operations.

Exploring Function Mapping

Function mapping describes how each element of a function's domain is paired with an element in its range. This concept is central to understanding how functions work and how to find their inverses. Consider a function \( f(x) \): it maps an input \( x \) from the domain to an output \( y \) in the codomain. In inverse function scenarios:

• If \( f(a) = b \), then the inverse function \( f^{-1}(b) = a \).
• The inverse function flips the mapping direction: output to input.

This clear mapping mechanism means that to find the inverse value for \( f \), one simply reverses this mapping. Thus, an understanding of function mapping is crucial, especially when dealing with one-to-one functions. It forms the groundwork for navigating through inverse tasks where the original and inverse operations redeploy their roles.

Inverse Operations in Functions

Inverse operations are like the undo button in mathematical functions. They are operations that reverse the effect of another operation. For functions, if you have \( f(x) \) and its inverse \( f^{-1}(x) \), applying them in succession gives you back your original input:

• When you apply \( f^{-1} \) after \( f \) to an element, you retrieve the original value: \( f^{-1}(f(x)) = x \).
• Similarly, applying \( f \) after \( f^{-1} \) recovers the original output: \( f(f^{-1}(x)) = x \).

In our exercise, this concept helps us find the missing values by effectively retracing our steps in the mappings provided. Through inverse operations, we directly achieve what the functions initially altered, thereby rendering inverse calculations straightforward and systematic.