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−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 .

GPT-4o mini · Answer

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 $x \geq 5$.