Problem 58 - Thomas Calculus

Question

If a composition \(f^{\circ} g\) is one-to-one, must \(g\) be one-to-one? Give reasons for your answer.

Step-by-Step Solution

Answer

Yes, if \(f^{\circ} g\) is one-to-one, then \(g\) must be one-to-one.

Step 1: Understand the Problem

We are asked to determine whether the function \(g\) must be one-to-one if the composition \(f^{\circ} g\) (i.e., \(f(g(x))\)) is one-to-one. This requires understanding what it means for a function to be one-to-one: a function \(h\) is one-to-one if, for every pair of different inputs, the outputs are different, i.e., \(h(a) eq h(b)\) whenever \(a eq b\).

Step 2: Recall Definitions

Recall that the composition of two functions \(f^{\circ} g\) is defined as \(f(g(x))\). A function is considered one-to-one (injective) if no two different elements in the domain map to the same element in the codomain.

Step 3: Analyze the Composition

Consider function \(f\) and \(g\), where the composition \(f(g(x))\) is one-to-one. If \(g(x_1) = g(x_2)\) for any \(x_1 eq x_2\), then \(f(g(x_1)) = f(g(x_2))\). This contradicts the assumption that \(f(g(x))\) is one-to-one. Thus, \(g\) must map different inputs to different outputs, suggesting \(g\) must be one-to-one.

Step 4: Conclude

Since we derived that \(g(x_1) = g(x_2)\) leads to a contradiction when \(f(g(x))\) is known to be one-to-one, we conclude that \(g\) must truly map distinct values to distinct outputs. Therefore, if \(f^{\circ} g\) is one-to-one, then \(g\) must be one-to-one.

Key Concepts

Composition of Functions

When we talk about the composition of functions, we are combining two functions to create a new function. If we have two functions, say \( f \) and \( g \), the composition is written as \( f \circ g \) or \( f(g(x)) \). This means that you first apply the function \( g \) to \( x \), and then apply the function \( f \) to the result.
The importance of the composition lies in how it combines processes: like first preparing ingredients with \( g \) before cooking them with \( f \). It is crucial to follow the order precisely for the desired outcome, just like following a recipe.
In mathematics, the order of operations can change the result, so understanding the sequence in function compositions is essential.

• In \( f(g(x)) \), always follow: apply \( g \) first, then \( f \).
• The composition reflects a chain, where the output of \( g \) becomes the input to \( f \).

This combination is not just theoretical; it's widely used in calculus, programming, and modeling real-world phenomena.

Injective Function

An injective function, also known as a one-to-one function, is a special type of function where each element in the function's domain is mapped to a unique element in the codomain.
In simple terms, different inputs will never map to the same output. This is a key feature in ensuring that a function's output is reliable and distinct.
You can visualize this by imagining a room full of unique keys (inputs) where each key fits into only one specific lock (output).

How to identify an injective function? Here are some pointers:

• If \( h(a) eq h(b) \) whenever \( a eq b \), then \( h \) is injective.
• No two different elements from the domain can have the same image in the codomain.

Injective functions are crucial in mathematics because they ensure distinctiveness in mapping, which is necessary for certain operations like inverse functions.

Function Mapping

Function mapping describes the relationship between a function’s input and its corresponding output. It serves as a blueprint for understanding how each element from one set relates to an element in another.
Think of a map guiding you from your house to school—it tells you exactly where you start and where you'll end up.

Here are some highlights of function mapping:

• It demonstrates how elements of the domain correspond to elements in the codomain.
• For a function \( f \colon A \rightarrow B \), every element in set \( A \) (domain) is paired with an element in set \( B \) (codomain).

Mapping is fundamental in understanding how functions operate because it visually or conceptually shows the relationship and journeys from inputs to outputs.

Function Domain

The domain of a function refers to all possible input values (or x-values) for which the function is defined. In simpler words, it's the set of all inputs that a function can handle without causing any errors or undefined behavior.

Why is the domain important?

• The domain sets the boundaries for which a function can operate correctly and logically.
• Understanding the domain prevents calculations from involving numbers or scenarios that a function can't handle.

For example, if a function involves a square root, the domain would include only non-negative numbers as inputs to keep the outputs real. Similarly, if the denominator in a fraction is a variable, the domain cannot include values that make the denominator zero.
Knowing the domain avoids mistakes and ensures that every input has a meaningful and valid output.