Inverse of composite functions


c. Explain why if g and h are one-to-one, the inverse of ƒ(x) = g(h(x)) exists.

Verified step by step guidance

1. To determine if the inverse of a composite function \( f(x) = g(h(x)) \) exists, we first need to understand the concept of one-to-one functions. A function is one-to-one (injective) if each output is mapped from a unique input, meaning no two different inputs produce the same output.
2. Given that both \( g \) and \( h \) are one-to-one functions, we can infer that \( h(x) \) maps each input \( x \) to a unique output \( y \), and then \( g(y) \) maps each \( y \) to a unique output \( z \). This ensures that the entire mapping from \( x \) to \( z \) through \( g(h(x)) \) is also one-to-one.
3. Since \( f(x) = g(h(x)) \) is a composition of two one-to-one functions, it is itself one-to-one. This is because the composition of two injective functions is injective.
4. For a function to have an inverse, it must be both one-to-one and onto (bijective). In this context, we are primarily concerned with the injective property, which is satisfied as shown.
5. Therefore, since \( f(x) = g(h(x)) \) is one-to-one, it has an inverse function \( f^{-1}(x) \), which can be found by reversing the operations of \( g \) and \( h \) in the reverse order, i.e., \( f^{-1}(x) = h^{-1}(g^{-1}(x)) \).

Key Concepts

One-to-One Functions

A function is considered one-to-one (injective) if it assigns distinct outputs to distinct inputs. This means that for any two different inputs, the outputs will also be different. One-to-one functions are crucial for the existence of inverses because they ensure that each output corresponds to exactly one input, preventing ambiguity in reversing the function.

Composite Functions

A composite function is formed when one function is applied to the result of another function, denoted as ƒ(x) = g(h(x)). The inner function h(x) is evaluated first, followed by the outer function g. Understanding composite functions is essential for analyzing the behavior of combined transformations and determining the conditions under which their inverses exist.

Existence of Inverses

For a function to have an inverse, it must be bijective, meaning it is both one-to-one and onto. In the case of composite functions, if both g and h are one-to-one, then their composition g(h(x)) will also be one-to-one. This guarantees that the inverse function exists, as each output from the composite function can be traced back to a unique input.