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Chapter 5: Problem 98

Is the converse of the second part of Theorem 5.7 true? That is, if a function is one-to-one (and therefore has an inverse function), then must the function be strictly monotonic? If so, prove it. If not, give a counterexample.

Short Answer

Expert verified
No, the converse of the second part of Theorem 5.7 is not true. A function can be one-to-one without being strictly monotonic. A counterexample is \(f(x) = x^3\), which is a one-to-one function but not strictly monotonic.

Step by step solution

01

Understanding the Concepts

Ensure you have a solid understanding of the terms 'strictly monotonic function', and 'one-to-one function' or 'injective function'. A function is said to be strictly monotonic if it is either consistently increasing or consistently decreasing. A function is said to be one-to-one, or injective, if it maps distinct input to distinct output.
02

Visualizing the Statement

Visualize the statement in question: 'If a function is one-to-one (and therefore has an inverse function), then must the function be strictly monotonic?'. This implies that every injective function must be strictly increasing or strictly decreasing.
03

Constructing a Counterexample

Let's consider a simple counterexample: the function \(f(x) = x^3\). For \(x = -1\), \(f(x) = -1\). For \(x = 0\), \(f(x) = 0\), and for \(x = 1\), \(f(x) = 1\). Here, \(f(x)\) is one-to-one (injective) as each unique input corresponds to a unique output. However, it is not strictly monotonic over its entire domain as it is not always increasing or always decreasing. It decreases from \(x = -1\) to \(x = 0\) and then increases from \(x = 0\) to \(x = 1\). So, \(f(x) = x^3\) is a valid counterexample that disproves the converse of theorem 5.7.

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