Advanced Calculus pp 151-184 | Cite as

# Inverses

## Abstract

Inverses help us solve equations: if 5 = *x*3, then \(x = \sqrt[3]{5}\). Equations also imply relations between their variables. For example, if *x*2 +*y*2 -1 = 0, then we can “solve for *y*” to get either \(y = + \sqrt {1 - x^2 }\) or \(y = - \sqrt {1 - x^2 }\). We soon learn that a formula for an inverse or for an implicitly defined function is seldom available. Usually, the most we can expect to know is that such a function exists. As we show, even this apparently limited knowledge can simplify and clarify our view of a problem, the same way that changing coordinates can simplify an integration. In this chapter, we look only briefly at explicit formulas. We give the bulk of our attention to the way inverses give us a powerful tool for understanding maps, and to the conditions that guarantee their existence. The next chapter does the same for implicitly defined functions.

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