QuestionAugust 25, 2025

Find (fcdot g)(x) and (g^circ f)(x) f(x)=-2x-1 g(x)=-6x-3 Write your answer as a polynomial in simplest form. (fcdot g)(x)=square (gcdot f)(x)=square

Find (fcdot g)(x) and (g^circ f)(x) f(x)=-2x-1 g(x)=-6x-3 Write your answer as a polynomial in simplest form. (fcdot g)(x)=square (gcdot f)(x)=square
Find (fcdot g)(x) and (g^circ f)(x) f(x)=-2x-1 g(x)=-6x-3 Write your answer as a polynomial in simplest form. (fcdot g)(x)=square (gcdot f)(x)=square

Solution
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Answer

(f \cdot g)(x) = 12x^2 + 12x + 3 ### (g^{\circ} f)(x) = 12x + 3 Explanation 1. Find (f \cdot g)(x) Multiply f(x) and g(x): (-2x-1)(-6x-3). 2. Expand the expression Use distribution: (-2x)(-6x) + (-2x)(-3) + (-1)(-6x) + (-1)(-3). 3. Simplify the terms Calculate: 12x^2 + 6x + 6x + 3. 4. Combine like terms Result: 12x^2 + 12x + 3. 5. Find (g^{\circ} f)(x) Substitute f(x) into g(x): g(f(x)) = g(-2x-1). 6. Evaluate g(-2x-1) Replace x in g(x) with -2x-1: -6(-2x-1) - 3. 7. Simplify the expression Calculate: 12x + 6 - 3. 8. Simplify further Result: 12x + 3.

Explanation

1. Find $(f \cdot g)(x)$<br /> Multiply $f(x)$ and $g(x)$: $(-2x-1)(-6x-3)$.<br />2. Expand the expression<br /> Use distribution: $(-2x)(-6x) + (-2x)(-3) + (-1)(-6x) + (-1)(-3)$.<br />3. Simplify the terms<br /> Calculate: $12x^2 + 6x + 6x + 3$.<br />4. Combine like terms<br /> Result: $12x^2 + 12x + 3$.<br /><br />5. Find $(g^{\circ} f)(x)$<br /> Substitute $f(x)$ into $g(x)$: $g(f(x)) = g(-2x-1)$.<br />6. Evaluate $g(-2x-1)$<br /> Replace $x$ in $g(x)$ with $-2x-1$: $-6(-2x-1) - 3$.<br />7. Simplify the expression<br /> Calculate: $12x + 6 - 3$.<br />8. Simplify further<br /> Result: $12x + 3$.
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