In Exercises 75-82, express the given function h as a composition of two functions ƒ and g so that h(x) = (fog) (x). h(x) = 1/(2x-3)

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this context, if h(x) = (f o g)(x), it means h(x) is the result of applying function g to x first, followed by applying function f to the result of g. Understanding how to manipulate and combine functions is essential for expressing h as a composition.

Rational Functions

A rational function is a function that can be expressed as the ratio of two polynomials. In the given function h(x) = 1/(2x-3), the denominator is a linear polynomial, which influences the behavior of the function, such as its domain and asymptotes. Recognizing the structure of rational functions is crucial for identifying suitable functions f and g.

Identifying Functions f and g

To express h(x) as a composition of two functions, one must identify appropriate functions f and g such that h(x) = f(g(x)). This often involves breaking down the original function into simpler components. For example, one might let g(x) be the inner function that transforms x, and f(x) be the outer function that processes the output of g, leading to the desired form of h.