Question

Use the functions below.

f(x)=−x−5−−−−√3
g(x)=2x−5−−−−√3
h(x)=x−5−−−−√3
What is (f⋅g⋅h)(x)
?

Enter the missing values.

(1 point)
(f⋅g⋅h)(x)=
$$

Answer 1

To find \((f \cdot g \cdot h)(x)\), we first need to multiply the three functions together.

Given:
- \( f(x) = -\sqrt[3]{x - 5} \)
- \( g(x) = 2\sqrt[3]{x - 5} \)
- \( h(x) = \sqrt[3]{x - 5} \)

Now, we will compute \((f \cdot g \cdot h)(x)\):

\[
(f \cdot g \cdot h)(x) = f(x) \cdot g(x) \cdot h(x)
\]

Substituting the expressions for \( f(x) \), \( g(x) \), and \( h(x) \):

\[
= (-\sqrt[3]{x - 5}) \cdot (2\sqrt[3]{x - 5}) \cdot (\sqrt[3]{x - 5})
\]

Next, combine the cube root terms:

\[
= -2(\sqrt[3]{x - 5}) \cdot (\sqrt[3]{x - 5}) \cdot (\sqrt[3]{x - 5})
\]

This is equivalent to:

\[
= -2(\sqrt[3]{(x - 5)(x - 5)(x - 5)}) = -2\sqrt[3]{(x - 5)^3}
\]

Since \(\sqrt[3]{(x - 5)^3} = x - 5\), we can simplify this further:

\[
= -2(x - 5)
\]

Therefore, we have:

\[
(f \cdot g \cdot h)(x) = -2(x - 5)
\]

So, the answer is:

\[
\boxed{-2(x - 5)}
\]