Question

Use the functions below.
ONLY USE GIVEN ANSWER.
f(x)=−2x2+4x−7
g(x)=−x−1−−−−√3+2
What is (f+g)(x)
? What is the domain of (f+g)(x)
?

(1 point)
Responses

(f+g)(x)=−2x2+4x−x−6−−−−−√3
and the domain of (f+g)(x)
is (−∞, ∞)
.
open paren f plus g close paren of x is equal to negative 2 x squared plus 4 x minus cube root of x minus 6 and the domain of open paren f plus g close paren of x is open paren negative infinity comma infinity close paren .

(f+g)(x)=−2x2+4x−x−1−−−−−√3−5
and the domain of (f+g)(x)
is (−∞, ∞)
.
open paren f plus g close paren of x is equal to negative 2 x squared plus 4 x minus cube root of x minus 1 minus 5 and the domain of open paren f plus g close paren of x is open paren negative infinity comma infinity close paren .

(f+g)(x)=−2x2+4x−x−6−−−−−√3
and the domain of (f+g)(x)
is x≥1
.
open paren f plus g close paren of x is equal to negative 2 x squared plus 4 x minus cube root of x minus 6 and the domain of open paren f plus g close paren of x is x is greater than or equal to 1 .

(f+g)(x)=−2x2+4x−x−1−−−−−√3−5
and the domain of (f+g)(x)
is x≥1
.
open paren f plus g close paren of x is equal to negative 2 x squared plus 4 x minus cube root of x minus 1 minus 5 and the domain of open paren f plus g close paren of x is x is greater than or equal to 1 .

Answer 1

To find \((f+g)(x)\), we start by combining the functions \(f(x)\) and \(g(x)\):

1. \(f(x) = -2x^2 + 4x - 7\)
2. \(g(x) = -\sqrt[3]{x-1} + 2\)

Now we can add them together:

\[
(f+g)(x) = f(x) + g(x) = (-2x^2 + 4x - 7) + (-\sqrt[3]{x-1} + 2)
\]

This simplifies to:

\[
(f+g)(x) = -2x^2 + 4x - 7 - \sqrt[3]{x-1} + 2 = -2x^2 + 4x - 5 - \sqrt[3]{x-1}
\]

Now regarding the domain:

The cube root function \(-\sqrt[3]{x-1}\) is defined for all real numbers, but there are no restrictions from the quadratic function \(-2x^2 + 4x - 5\) either. Hence, the domain is all real numbers:

So, we have:

\[
(f+g)(x) = -2x^2 + 4x - 5 - \sqrt[3]{x-1}
\]
and the domain of \((f+g)(x)\) is \((-\infty, \infty)\).

Based on the provided options, the response that matches our calculation is:

(f+g)(x)=−2x^2+4x−x−1−−−−−√3−5 and the domain of (f+g)(x) is (−∞, ∞).