Question

homework

1.) given that \\(f(x) = 2x + 1\\), \\(g(x) = x^2\\)
find \\((f \cdot g)(x)\\) b) \\((g \cdot f)(x)\\)

2.) if \\(f(x) = 3x - 4\\) and \\(g(x) = x + 2\\) find \\((f \cdot g)(5)\\)

Explanation:

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<reasoning>

Find the product function (f * g)(x)

\[
(f \cdot g)(x) = f(x) \cdot g(x) = (2x + 1)(x^2) = 2x^3 + x^2
\]

Find the composite function (g o f)(x)

\[
(g \circ f)(x) = g(f(x)) = (2x + 1)^2 = 4x^2 + 4x + 1
\]

Evaluate the product function (f * g)(5)

\[

$$\begin{aligned} f(5) &= 3(5) - 4 = 11\\ g(5) &= 5 + 2 = 7\\ (f \cdot g)(5) &= f(5) \cdot g(5) = 11 \cdot 7 = 77 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

a) \((f \cdot g)(x) = 2x^3 + x^2\)
b) \((g \circ f)(x) = 4x^2 + 4x + 1\)

Question 2

\((f \cdot g)(5) = 77\)
</answer>

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Answer:

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<reasoning>

Find the product function (f * g)(x)

\[
(f \cdot g)(x) = f(x) \cdot g(x) = (2x + 1)(x^2) = 2x^3 + x^2
\]

Find the composite function (g o f)(x)

\[
(g \circ f)(x) = g(f(x)) = (2x + 1)^2 = 4x^2 + 4x + 1
\]

Evaluate the product function (f * g)(5)

\[

$$\begin{aligned} f(5) &= 3(5) - 4 = 11\\ g(5) &= 5 + 2 = 7\\ (f \cdot g)(5) &= f(5) \cdot g(5) = 11 \cdot 7 = 77 \end{aligned}$$

\]
</reasoning>

<answer>

Question 1

a) \((f \cdot g)(x) = 2x^3 + x^2\)
b) \((g \circ f)(x) = 4x^2 + 4x + 1\)

Question 2

\((f \cdot g)(5) = 77\)
</answer>

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