Find fg and determine the domain for each function. f(x) = x -5, g(x) = 3x²

Verified step by step guidance

1. Step 1: Understand the problem. You are tasked with finding the composition of two functions, denoted as fg(x), which means f(g(x)). This involves substituting the entire function g(x) into the function f(x). Additionally, you need to determine the domain of the resulting composition function.
2. Step 2: Write the given functions. The functions provided are f(x) = x - 5 and g(x) = 3x². To find fg(x), substitute g(x) into f(x). This means replacing every instance of x in f(x) with the expression for g(x).
3. Step 3: Perform the substitution. Replace x in f(x) = x - 5 with g(x) = 3x². This gives f(g(x)) = (3x²) - 5. Simplify the expression to get the composition function fg(x).
4. Step 4: Determine the domain of fg(x). The domain of a function is the set of all x-values for which the function is defined. For fg(x) = 3x² - 5, there are no restrictions such as square roots or denominators, so the domain is all real numbers, which can be written as (-∞, ∞).
5. Step 5: Summarize the results. The composition function fg(x) is f(g(x)) = 3x² - 5, and its domain is all real numbers, (-∞, ∞).

Key Concepts

Function Composition

Function composition involves combining two functions, where the output of one function becomes the input of another. In this case, fg means f(g(x)), which requires substituting g(x) into f(x). Understanding how to perform this substitution is crucial for finding the composed function.

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. When composing functions, the domain of the resulting function fg is determined by the domain of g(x) and any restrictions imposed by f(x). Identifying these domains ensures that the composed function is valid.

Quadratic Functions

Quadratic functions are polynomial functions of degree two, typically expressed in the form g(x) = ax² + bx + c. In this problem, g(x) = 3x² is a quadratic function, and understanding its properties, such as its shape (a parabola) and its domain (all real numbers), is essential for analyzing the composition with f(x).