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Chapter 1: Problem 2

Give an example of a function that is one-to-one on the entire real number line.

Short Answer

Expert verified
Question: Provide an example of a function that is one-to-one (injective) on the entire real number line. Answer: f(x) = x, the identity function is one-to-one on the entire real number line.

Step by step solution

01

Choose a function

Let's consider a linear function, f(x) = ax + b, where a and b are constants. A linear function may or may not be one-to-one, depending on the values of its constants. In this step, we will find the values for a and b that make the function one-to-one. #step2#
02

Ensure that f(x) = f(y) implies x = y

For our linear function to be one-to-one, we need to ensure that if f(x) = f(y), then x = y. If f(x) = ax + b and f(y) = ay + b, then setting f(x) = f(y) gives ax + b = ay + b. We now need to find the relationship between a and b that satisfies x = y under this condition. #step3#
03

Solve for the value of a

From the previous step, we have ax + b = ay + b. To find the values of a and b that make this equation true, we can subtract b from both sides and then solve for a: (ax + b) - b = (ay + b) - b This simplifies to: ax = ay Now divide both sides by x (assuming x 鈮� y and x 鈮� 0, since if x = y, the one-to-one property is true, and if x = 0, the dividing is not possible): a = ay/x Now, the only possible a value to make this equation true for all non-zero x and y (x 鈮� y) is a = 1. #step4#
04

Choose the value of b

Since our function is f(x) = ax + b and we have found a = 1, our function is now f(x) = x + b. The value of b can be any real number, as it just shifts the function up or down and does not affect the one-to-one property. For simplicity, let's choose b = 0. #step5#
05

Write the final function

Now we have determined the values for a and b that make our linear function one-to-one on the entire real number line. The final function is: f(x) = x This is a simple identity function and is one-to-one on the entire real number line, as every input has a unique output.

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Most popular questions from this chapter

Sketch a graph of the given pair of functions to conjecture a relationship between the two functions. Then verify the conjecture. $$\sin ^{-1} x ; \frac{\pi}{2}-\cos ^{-1} x$$

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