Let f(x)=x^2 and g(x)=1 / x . Compute f[g(x)] and g[f(x)] and note that the results are identica

step 1 They want us to find the composition of f composed with g as well as g composed with f, and then

Let f(x)=x^2 and g(x)=1 / x . Compute f[g(x)] and g[f(x)]...

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Key Concepts

Function Composition

Function composition involves creating a new function by applying one function to the results of another. In this context, the operations f(g(x)) and g(f(x)) illustrate how the output of one function becomes the input to the other, which is central to understanding how functions can be combined and potentially inverted.

Inverse Functions

Inverse functions are pairs of functions that 'undo' each other, meaning that when one function is applied after the other, the result is the original input. This requires that the compositions f(g(x)) and g(f(x)) return x for all x in the domain. The failure of f(x)=x² and g(x)=1/x to satisfy these conditions over their complete domains illustrates why they are not true inverses.

Domain and One-to-One Properties

For a function to have an inverse, it must be one-to-one (injective) and its domain must be appropriate for the inverse operation. Although f(g(x)) and g(f(x)) might appear identical, f(x)=x² is not one-to-one over the set of all real numbers because different inputs can produce the same output. This lack of injectivity, coupled with the fact that their domains do not align correctly for inverse relationships, explains why these functions do not qualify as inverses of each other.

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