In Exercises 31–50, find fg and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 17

Verified step by step guidance

1. First, understand that fg represents the product of the two functions f(x) and g(x). Therefore, fg(x) = f(x) * g(x).
2. Substitute the given functions into the expression: fg(x) = (3 - x^2) * (x^2 + 2x - 17).
3. Expand the expression by distributing each term in the first polynomial (3 - x^2) across each term in the second polynomial (x^2 + 2x - 17).
4. Combine like terms to simplify the expression into a single polynomial.
5. Determine the domain of fg(x). Since both f(x) and g(x) are polynomials, their domain is all real numbers. Therefore, the domain of fg(x) is also 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 composite 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 finding the domain of the composite function fg, it is essential to consider the domains of both f(x) and g(x) and ensure that the output of g(x) falls within the domain of f(x).

Quadratic Functions

Both f(x) and g(x) are quadratic functions, which are polynomial functions of degree two. The general form of a quadratic function is ax² + bx + c. Understanding the properties of quadratic functions, such as their parabolas' shapes and vertex, is important for analyzing their behavior and determining their domains.