For the pair of functions defined, find (ƒg)(x). Give the domain of each. See Example 2. ƒ(x)=2x^2-3x, g(x)=x^2-x+3

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. In this case, (ƒg)(x) means substituting g(x) into ƒ(x). This process requires evaluating the outer function ƒ at the output of the inner function g, which can lead to a new expression that represents the combined behavior of both functions.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For polynomial functions like ƒ(x) and g(x), the domain is typically all real numbers. However, when composing functions, it is essential to consider the domain of the inner function g(x) and ensure that its output falls within the domain of the outer function ƒ(x).

Polynomial Functions

Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The functions given, ƒ(x) = 2x² - 3x and g(x) = x² - x + 3, are both polynomials. Understanding their structure helps in analyzing their behavior, such as finding their composition and determining their domains.