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Chapter 8: Problem 90

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The function \(f(x)=5\) is one-to-one.

Short Answer

Expert verified
False. To make it true, we should state: The function \(f(x) = 5\) is not one-to-one.

Step by step solution

01

Understand the Definition of a One-to-One Function

A function is one-to-one (or injective) only if every y-value has exactly one x-value associated with it, that is, no two different inputs \(x_1\) and \(x_2\) produce the same output such that \(f(x_1) 鈮� f(x_2)\) for all \(x_1 鈮� x_2\).
02

Determine the outputs of the function \(f(x) = 5\)

The given function \(f(x) = 5\) is a constant function, meaning that for any input x, the output will always be 5.
03

Check if the Function is One-to-One

By comparing our understanding of one-to-one functions with the outputs of the function \(f(x) = 5\), it's clear that different inputs x, will not result in different y-values. Rather all inputs x result in the same output, 5. Therefore, we can conclude that function \(f(x) = 5\) is not a one-to-one function.
04

Correct false statement to make it true

As the statement 'The function \(f(x) = 5\) is one-to-one' is false, to make it true, we need to change it to: 'The function \(f(x) = 5\) is not one-to-one'.

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Most popular questions from this chapter

\(f\) and \(g\) are defined by the following tables. Use the tables to evaluate each composite function. $$\begin{array}{c|c}\hline x & f(x) \\\\\hline-1 & 1 \\\\\hline 0 & 4 \\\\\hline 1 & 5 \\\\\hline 2 & -1 \\ \hline\end{array}$$ $$\begin{array}{c|c}\hline x & g(x) \\\\\hline-1 & 0 \\\\\hline 1 & 1 \\\\\hline 4 & 2 \\\\\hline 10 & -1 \\ \hline\end{array}$$ $$f(g(4))$$

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Solve: \(3(6-x)=3-2(x-4)\).

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Find \(f(g(x))\) and \(g(f(x))\) and determine whether each pair of functions \(f\) and \(g\) are inverses of each other. $$f(x)=\frac{2}{x-5} \quad \text { and } \quad g(x)=\frac{2}{x}+5$$
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