Problem 18 - Algebra and Trigonometry

Question

Assume \(f\) is a one-to-one function. (a) If \(f(5)=18,\) find \(f^{-1}(18)\) (b) If \(f^{-1}(4)=2,\) find \(f(2)\)

Step-by-Step Solution

Answer

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

Step 1: Understanding the Problem

We are tasked with finding the inverse of the function values. In a one-to-one function, the inverse function reverses the effect of the original function.

Step 2: Solving Part (a)

Given that \(f(5) = 18\), to find \(f^{-1}(18)\), apply the property of inverse functions: \(f^{-1}(f(x)) = x\). Here, \(x = 5\) when \(f(x) = 18\). Thus, \(f^{-1}(18) = 5\).

Step 3: Solving Part (b)

Given that \(f^{-1}(4) = 2\), this means that \(f(2) = 4\). This again uses the inverse property \(f(f^{-1}(y)) = y\), showing \(f(2) = 4\).

Key Concepts

One-to-One Functions

A one-to-one function, also known as an injective function, is a type of function where each output value is associated with exactly one input value. This unique pairing ensures that the function is invertible, meaning an inverse function can be defined. Essentially, in a one-to-one function, if you have two different inputs, you will always have two different outputs. This allows us to easily "reverse" the function using its inverse.

Some important characteristics of one-to-one functions include:

• Every element of the range corresponds to exactly one element of the domain.
• No two different inputs map to the same output.
• Graphically, these functions pass the Horizontal Line Test, meaning any horizontal line drawn through the graph intersects the graph at most once.

Given these properties, one-to-one functions are particularly useful in solving equations and understanding inverse functions, as they guarantee unique solutions.

Function Properties

Functions have several key properties that help us understand how they behave and how we can manipulate them. When dealing with inverse functions and one-to-one functions, there are specific properties to focus on.First is the existence of an inverse. For a function to have an inverse, it must be one-to-one. This ensures that each output has a unique input, allowing us to reverse the direction of the function. For instance, if a function's property indicates that it is one-to-one as mentioned in the original exercise, finding inverses becomes straightforward.

Additionally, we observe the identity property, which essentially states that applying a function and its inverse in succession returns the original value:

• For a function \( f \) and its inverse \( f^{-1} \), the identity properties are: \( f(f^{-1}(y)) = y \) and \( f^{-1}(f(x)) = x \).

Understanding these properties allows us to solve problems like the original exercise where given values from the function or its inverse are used to find corresponding elements.

Function Notation

Function notation is a way of representing functions in a standard format, using symbols such as \( f(x) \) to denote a function \( f \) of \( x \). This notation plays a crucial role in communicating mathematical ideas clearly and efficiently.Here's a brief explanation of key components of function notation:

• \( f \) is the name of the function. It's a placeholder and can be replaced with any letter to denote different functions, like \( g \) or \( h \).
• \( x \) is the variable representing an element from the domain of the function. It is the input value for which the function provides an output.
• \( f(x) \) denotes the output of the function \( f \) when \( x \) is the input, often called the "value of \( f \) at \( x \)."

In terms of inverse functions, the notation \( f^{-1} \) is employed to indicate the inverse of the function \( f \). For example, \( f^{-1}(y) \) represents the input that gives the output \( y \) under the original function \( f \). Proper understanding of function notation simplifies solving inverse function problems, like those provided in the step-by-step solution.