Find fg and determine the domain for each function. f(x) = 3 − x², g(x) = x² + 2x − 15

검증된 단계별 안내

Step 1: Understand the problem. We are tasked with finding the composition of two functions, fg(x), which means f(g(x)). This involves substituting the entire expression for g(x) into the function f(x).

Step 2: Write the composition fg(x). Substitute g(x) = x² + 2x − 17 into f(x) = 3 − x². This gives f(g(x)) = 3 − (g(x))². Replace g(x) with its expression to get f(g(x)) = 3 − (x² + 2x − 17)².

Step 3: Simplify the expression for fg(x). Expand the squared term (x² + 2x − 17)² using the formula (a + b + c)² = a² + 2ab + 2ac + b² + 2bc + c². Then simplify the resulting polynomial.

Step 4: Determine the domain of fg(x). The domain of fg(x) is determined by the domain of g(x) and any restrictions introduced by the composition. Since g(x) = x² + 2x − 17 is a polynomial, it has no restrictions. However, check if any restrictions arise from the square in f(g(x)).

Step 5: Conclude the domain. After verifying there are no restrictions (e.g., no square roots or divisions by zero), the domain of fg(x) is all real numbers, which can be written as (-∞, ∞).

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주요 개념

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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 must consider the domains of both f and g, ensuring 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 is ax² + bx + c, where a, b, and c are constants. Understanding the properties of quadratic functions, such as their parabolas' shapes and vertex, is essential for analyzing their behavior and determining their domains.