Question

Let lower f left-parenthesis x right-parenthesis equals Start Root x minus 2 End Root and lower g left-parenthesis x right-parenthesis equals Start Root x plus 7 End Root. Find left-parenthesis f dot g right-parenthesis left-parenthesis x right-parenthesis. Assume all appropriate restrictions to the domain.
(1 point)
Responses

left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 5 x minus 14
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 5 x minus 14

left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 9 x minus 14
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals x squared plus 9 x minus 14

left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 9 x minus 14 End Root
Image with alt text: left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 9 x minus 14 End Root

left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 5 x minus 14 End Root

Answer 1

To find \((f \cdot g)(x)\), we first need to find the expressions for \(f(x)\) and \(g(x)\):

- \(f(x) = \sqrt{x - 2}\)
- \(g(x) = \sqrt{x + 7}\)

Now, we can compute the product:

\[
(f \cdot g)(x) = f(x) \cdot g(x) = \sqrt{x - 2} \cdot \sqrt{x + 7}
\]

Using the property of square roots that \(\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}\), we have:

\[
(f \cdot g)(x) = \sqrt{(x - 2)(x + 7)}
\]

Next, we need to simplify \((x - 2)(x + 7)\):

\[
(x - 2)(x + 7) = x^2 + 7x - 2x - 14 = x^2 + 5x - 14
\]

So we can express \((f \cdot g)(x)\) as:

\[
(f \cdot g)(x) = \sqrt{x^2 + 5x - 14}
\]

Considering the domain restrictions for \(f(x)\) and \(g(x)\):

1. For \(f(x) = \sqrt{x - 2}\), the expression inside the square root must be non-negative, so \(x - 2 \geq 0\) implies \(x \geq 2\).
2. For \(g(x) = \sqrt{x + 7}\), the expression inside the square root must be non-negative, so \(x + 7 \geq 0\) implies \(x \geq -7\).

However, since the more restrictive condition is \(x \geq 2\), that's the domain we will consider.

Thus, the final result is:

\[
(f \cdot g)(x) = \sqrt{x^2 + 5x - 14}
\]

The matching response is:

\[
\text{left-parenthesis lower f dot lower g right-parenthesis left-parenthesis x right-parenthesis equals Start Root x squared plus 5 x minus 14 End Root}
\]