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)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to express the given function h(x) = 1/(2x - 3) as a composition of two functions f(x) and g(x) such that h(x) = f(g(x)).
2. Step 2: Identify the inner function g(x). Look at the expression inside the denominator of h(x), which is (2x - 3). Let g(x) = 2x - 3.
3. Step 3: Identify the outer function f(x). After substituting g(x) into h(x), the remaining operation is taking the reciprocal of g(x). Let f(x) = 1/x.
4. Step 4: Verify the composition. Substitute g(x) into f(x): f(g(x)) = f(2x - 3) = 1/(2x - 3), which matches h(x).
5. Step 5: Conclude that the functions are f(x) = 1/x and g(x) = 2x - 3, and their composition satisfies h(x) = f(g(x)).

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. If we have two functions f(x) and g(x), the composition is denoted as (f o g)(x) = f(g(x)). Understanding this concept is crucial for expressing a function as a composition of two simpler functions.

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. Recognizing the structure of rational functions helps in identifying suitable functions f and g for composition.

Identifying Functions for Composition

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 manipulating the original function to isolate components that can be defined as g(x) and then determining f(x) based on g(x). This skill is essential for solving the problem effectively.