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Chapter 13: Problem 9

Do all functions have inverses? Explain your answer.

Short Answer

Expert verified
Not all functions have inverses. A function must be both injective (one-to-one) and surjective (onto) to have an inverse; such functions are called bijective functions. For example, the function \(f(x) = 3x - 2\) is bijective and has an inverse \(f^{-1}(x) = \frac{x+2}{3}\). However, the function \(g(x) = x^2\) does not have an inverse since it is not injective. Only bijective functions have inverses.

Step by step solution

01

Understanding Inverses

An inverse function is a function that reverses the input-output relationship established by the original function. Given a function \(f(x)\), its inverse is represented by \(f^{-1}(x)\) such that if \(y = f(x)\) then \(x = f^{-1}(y)\). For a function to have an inverse, it needs to satisfy two conditions: it must be both injective (one-to-one) and surjective (onto).
02

Examining Injectivity

A function is injective if every element in the domain is mapped to a unique element in the codomain and no two elements in the domain have the same image. In other words, if \(f(x_1) = f(x_2)\), then \(x_1 = x_2\). This ensures that the inverse function maps each output uniquely back to the corresponding input.
03

Examining Surjectivity

A function is surjective if every element in the codomain has at least one element in the domain mapped to it. This ensures that the inverse function can map to every element in the domain of the original function.
04

Identifying Invertible Functions

A function is invertible if it is both injective and surjective. Such functions are called bijective functions. In other words, for a function to have an inverse, it must be bijective.
05

Providing Examples

An example of a function with an inverse is \(f(x) = 3x - 2\). This function is injective because for any two inputs \(x_1\) and \(x_2\) with \(f(x_1) = f(x_2)\), we have \(3x_1 - 2 = 3x_2 - 2\) which leads to \(x_1 = x_2\). It is also surjective, since any real number can be reached in the codomain by choosing the corresponding real number in the domain. Hence, this function is bijective and has an inverse \(f^{-1}(x) = \frac{x+2}{3}\). On the other hand, a function without an inverse is \(g(x) = x^2\). While it is surjective for the non-negative real numbers, it is not injective as \(g(x_1) = g(x_2)\) for \(x_1 = -x_2\). Therefore, \(g(x)\) does not have an inverse.
06

Conclusion

Based on the given examples and the criteria for a function to have an inverse, we can conclude that not all functions have inverses. Only bijective functions have inverses.

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