Use the functions below.

Question

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−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 .

(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−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−5−−−−−√−5
and the domain of (f+g+h)(x)
is (−∞, ∞)
.

Answer 1

To calculate \((f+g+h)(x)\), we will first define each function explicitly and then sum them:

1. Define the functions:

   • \( f(x) = \sqrt{x - 5} + 2 \)
   • \( g(x) = -2\sqrt{x - 5} - 7 \)
   • \( h(x) = -\sqrt{x - 5} \)

2. Sum the functions: \[ (f + g + h)(x) = f(x) + g(x) + h(x) \] \[ = (\sqrt{x - 5} + 2) + (-2\sqrt{x - 5} - 7) + (-\sqrt{x - 5}) \] Combining like terms: \[ = \sqrt{x - 5} - 2\sqrt{x - 5} - \sqrt{x - 5} + 2 - 7 \] The square root terms combine to: \[ = (1 - 2 - 1)\sqrt{x - 5} + (2 - 7) = -2\sqrt{x - 5} - 5 \]

Thus, we have: \[ (f + g + h)(x) = -2\sqrt{x - 5} - 5 \]

3. Determine the domain: The domain of each function involves the term \(\sqrt{x - 5}\). The expression under the square root must be non-negative: \[ x - 5 \geq 0 \ x \geq 5 \]

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

Conclusion:

• \((f + g + h)(x) = -2\sqrt{x - 5} - 5\)
• The domain of \((f + g + h)(x)\) is \( x \geq 5 \).

The correct response based on your options would be: \[ (f + g + h)(x) = -2\sqrt{x - 5} - 5 \text{ and the domain of } (f + g + h)(x) \text{ is } x \geq 5. \]