Math Problem Statement

Find \((f \cdot g)(x)\) if \(f(x) = x + 2\) and \(g(x) = x^2 + 4x + 4\).

Solution

To solve for (fg)(x)(f \cdot g)(x) when f(x)=x+2f(x) = x + 2 and g(x)=x2+4x+4g(x) = x^2 + 4x + 4, we need to calculate the product of f(x)f(x) and g(x)g(x), which is defined as:

(fg)(x)=f(x)g(x)(f \cdot g)(x) = f(x) \cdot g(x)

Step-by-Step Solution

  1. Write down f(x)f(x) and g(x)g(x):

    • f(x)=x+2f(x) = x + 2
    • g(x)=x2+4x+4g(x) = x^2 + 4x + 4
  2. Multiply f(x)f(x) and g(x)g(x): (fg)(x)=(x+2)(x2+4x+4)(f \cdot g)(x) = (x + 2)(x^2 + 4x + 4)

  3. Expand the expression:

    • Distribute xx through (x2+4x+4)(x^2 + 4x + 4): x(x2+4x+4)=x3+4x2+4xx \cdot (x^2 + 4x + 4) = x^3 + 4x^2 + 4x
    • Distribute 22 through ((x^2

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Math Problem Analysis

Mathematical Concepts

Algebra
Polynomial Functions
Function Multiplication

Formulas

\((f \cdot g)(x) = f(x) \cdot g(x)\)
Polynomial expansion

Theorems

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Suitable Grade Level

Grades 9-12