Questions: f(x)=x+3 g(x)=2x^2-4 Find (f cdot g)(x). A. (f cdot g)(x)=2x^3-12 B. (f cdot g)(x)=2x^3+6x^2+4x+12 C. (f cdot g)(x)=2x^3+6x^2-4x-12 D. (f cdot g)(x)=2x^3+12

Transcript text: \[ \begin{array}{l} f(x)=x+3 \\ g(x)=2 x^{2}-4 \end{array} \] Find $(f \cdot g)(x)$. A. $(f \cdot g)(x)=2 x^{3}-12$ B. $(f \cdot g)(x)=2 x^{3}+6 x^{2}+4 x+12$ c. $(f \cdot g)(x)=2 x^{3}+6 x^{2}-4 x-12$ D. $(f: g)(x)=2 x^{3}+12$

Solution

Solution Steps

Step 1: Understand the Problem

We are given two functions: \[ f(x) = x + 3 \] \[ g(x) = 2x^2 - 4 \]

We need to find the product of these two functions, denoted as $(f \cdot g)(x)$.

Step 2: Multiply the Functions

To find $(f \cdot g)(x)$, we multiply $f(x)$ by $g(x)$: \[ (f \cdot g)(x) = (x + 3)(2x^2 - 4) \]

Step 3: Distribute the Terms

Distribute each term in $f(x)$ across each term in $g(x)$: \[ = x(2x^2 - 4) + 3(2x^2 - 4) \]

Step 4: Simplify the Expression

Calculate each part: \[ = (x \cdot 2x^2) + (x \cdot -4) + (3 \cdot 2x^2) + (3 \cdot -4) \] \[ = 2x^3 - 4x + 6x^2 - 12 \]

Reorder the terms: \[ = 2x^3 + 6x^2 - 4x - 12 \]

Final Answer

The correct expression for $(f \cdot g)(x)$ is: \[ \boxed{2x^3 + 6x^2 - 4x - 12} \]

Thus, the answer is C.

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