Problem 8 - Precalculus : Building Concepts and Connections

Question

\(f\) and \(g\) are inverses of each other. True or False? \(f\) is a one-to-one function.

Step-by-Step Solution

Answer

True. If a function has an inverse, it must be one-to-one.

Step 1: Understanding the Statement

Recognize that \(f\) and \(g\) are inverses of each other. This means \(f(g(x)) = x\) and \(g(f(x)) = x\) for every x in their domains.

Step 2: Understanding one-to-one

Understand that a one-to-one function is a function for which every unique input corresponds to a unique output. This means if \(f(a) = f(b)\) then a = b.

Step 3: Inverse of a Function implies One-to-One

Note that for a function to have an inverse, it must be one-to-one. Because \(f\) has an inverse \(g\), it must be the case that \(f\) is one-to-one.

Key Concepts

One-to-One Function

A one-to-one function is a special type of function where each element in the domain maps to a unique element in the codomain. This characteristic ensures that no two distinct elements in the domain will ever map to the same element in the range. In mathematical terms, if a function \( f \) is one-to-one, then whenever \( f(a) = f(b) \), it must be true that \( a = b \). Here are some important points about one-to-one functions:

• If you plot a one-to-one function on a graph, it will pass the horizontal line test. This means that any horizontal line will intersect the graph at most once.
• One-to-one functions are essential because only these types of functions can have inverses. This inverse function will reverse the mapping of the original function, ensuring that for every output there is a unique input.

Understanding one-to-one functions helps in grasping the concept of inverse functions. Since the exercise mentioned that \( f \) and \( g \) are inverses, it follows that \( f \) must be one-to-one.

Function Composition

Function composition involves combining two functions such that the output of one function becomes the input of another. When dealing with two functions \( f \) and \( g \), the composition is denoted as \( f(g(x)) \), which means you first apply \( g \) to \( x \) and then apply \( f \) to the result of \( g(x) \). Let's explore some key points about function composition:

• The composition of functions \( f \) and \( g \) is not generally commutative. This means that generally \( f(g(x)) \) is not equal to \( g(f(x)) \).
• This concept is vital in understanding inverse functions. For \( f \) and \( g \) to be inverses, both \( f(g(x)) = x \) and \( g(f(x)) = x \) must hold true for every \( x \) in their respective domains.

Function composition is an effective method for verifying if two functions are indeed inverses, as it highlights the function's behavior in relation to its outputs.

Domain of a Function

The domain of a function is defined as the complete set of possible input values (or 'x' values) for which the function is defined. Understanding the domain is crucial when dealing with functions and especially when checking their inverses. This is because the function can only be inverse when it is bijective, meaning both one-to-one and onto, across its domain. Important aspects concerning the domain:

• The domain must be considered when working with composed functions like \( f(g(x)) \) or \( g(f(x)) \). Both functions need to accept the inputs from one another.
• Sometimes, the domain of an inverse function can be different from the domain of the original function. The domain of the inverse is actually the range of the original function.

Understanding the domain of a function aids in properly framing the function’s inverses and ensures that when we reverse the outputs back to inputs, they fit within the allowed set of numbers.