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 and composition functions for Question 1

\[

$$\begin{aligned} (f \cdot g)(x) &= f(x) \cdot g(x) = (2x + 1)(x^2) = 2x^3 + x^2\\ (g \cdot f)(x) &= g(x) \cdot f(x) = (x^2)(2x + 1) = 2x^3 + x^2 \end{aligned}$$

\]

Evaluate the product function at x = 5 for Question 2

\[

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

\]
</reasoning>

<answer>

Question 1

\[

$$\begin{aligned} (f \cdot g)(x) &= 2x^3 + x^2\\ (g \cdot f)(x) &= 2x^3 + x^2 \end{aligned}$$

\]

Question 2

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

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Answer:

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<reasoning>

Find the product and composition functions for Question 1

\[

$$\begin{aligned} (f \cdot g)(x) &= f(x) \cdot g(x) = (2x + 1)(x^2) = 2x^3 + x^2\\ (g \cdot f)(x) &= g(x) \cdot f(x) = (x^2)(2x + 1) = 2x^3 + x^2 \end{aligned}$$

\]

Evaluate the product function at x = 5 for Question 2

\[

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

\]
</reasoning>

<answer>

Question 1

\[

$$\begin{aligned} (f \cdot g)(x) &= 2x^3 + x^2\\ (g \cdot f)(x) &= 2x^3 + x^2 \end{aligned}$$

\]

Question 2

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

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