Question

Use the functions below.

f(x)=x−5−−−−√+2
g(x)=−2x−5−−−−√−7
h(x)=−x−5−−−−√
What is (f+g+h)(x)
? What is the domain of (f+g+h)(x)
?

(1 point)
Responses

(f+g+h)(x)=−2x−10−−−−−√
and the domain of (f+g+h)(x)
is (−∞, ∞)
.
open paren f plus g plus h close paren times x is equal to negative 2 square root of x minus 10 end root and the domain of open paren f plus g plus h close paren times x is open paren negative infinity comma infinity close paren .

(f+g+h)(x)=−2x−10−−−−−√
and the domain of (f+g+h)(x)
is x≥10
.
open paren f plus g plus h close paren times x is equal to negative 2 square root of x minus 10 end root and the domain of open paren f plus g plus h close paren times x is x is greater than or equal to 10 .

(f+g+h)(x)=−2x−5−−−−−√−5
and the domain of (f+g+h)(x)
is (−∞, ∞)
.
open paren f plus g plus h close paren times x is equal to negative 2 square root of x minus 5 end root minus 5 and the domain of open paren f plus g plus h close paren times x is open paren negative infinity comma infinity close paren .

(f+g+h)(x)=−2x−5−−−−−√−5
and the domain of (f+g+h)(x)
is x≥5
.
open paren f plus g plus h close paren times x is equal to negative 2 square root of x minus 5 end root minus 5 and the domain of open paren f plus g plus h close paren times x is x is greater than or equal to 5 .

Answer 1

To find \((f + g + h)(x)\), we need to combine the given functions \(f(x)\), \(g(x)\), and \(h(x)\).

1. Given:
- \(f(x) = \sqrt{x - 5} + 2\)
- \(g(x) = -2\sqrt{x - 5} - 7\)
- \(h(x) = -\sqrt{x - 5}\)

2. Let's compute \((f + g + h)(x)\):

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

Plugging in the expressions:

\[
(f + g + h)(x) = \left(\sqrt{x - 5} + 2\right) + \left(-2\sqrt{x - 5} - 7\right) + \left(-\sqrt{x - 5}\right)
\]

Combining like terms:

\[
= \sqrt{x - 5} - 2\sqrt{x - 5} - \sqrt{x - 5} + 2 - 7
\]
\[
= (1 - 2 - 1)\sqrt{x - 5} + (2 - 7)
\]
\[
= -2\sqrt{x - 5} - 5
\]

Therefore, we have:

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

3. Next, we need to determine the domain of \((f + g + h)(x)\). The domain of the combined function is influenced by the domain of each individual function.

- For \(f(x)\), \(\sqrt{x - 5}\) requires \(x - 5 \geq 0 \Rightarrow x \geq 5\).
- For \(g(x)\), \(-2\sqrt{x - 5}\) also requires \(x - 5 \geq 0 \Rightarrow x \geq 5\).
- For \(h(x)\), \(-\sqrt{x - 5}\) requires \(x - 5 \geq 0 \Rightarrow x \geq 5\).

Therefore, the domain of \((f + g + h)(x)\) is \(x \geq 5\).

Putting this all together, we find:

\[
(f + g + h)(x) = -2\sqrt{x - 5} - 5
\]
and the domain is \(x \geq 5\).

**Final answer:**
- \((f+g+h)(x) = -2\sqrt{x-5} - 5\)
- The domain is \(x \geq 5\).