Question

Use the functions below.

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−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 (−∞, ∞)
.
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−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 .
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Answer 1

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

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

2. **Add the functions:**
\[
(f + g)(x) = f(x) + g(x) = (-2x^2 + 4x - 7) + (-\sqrt[3]{x - 1} + 2)
\]
Simplifying this:
\[
(f + g)(x) = -2x^2 + 4x - 7 - \sqrt[3]{x - 1} + 2
\]
\[
(f + g)(x) = -2x^2 + 4x - 5 - \sqrt[3]{x - 1}
\]

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

3. **Determine the domain of \((f + g)(x)\):**
- The function \(f(x)\) is a polynomial, which is defined for all \(x\).
- The function \(g(x) = -\sqrt[3]{x - 1} + 2\) is also defined for all \(x\) since the cube root function is defined for all real numbers.

Thus, the domain of \((f + g)(x)\) is:
\[
(-\infty, \infty)
\]

### Conclusion:
- \((f + g)(x) = -2x^2 + 4x - 5 - \sqrt[3]{x - 1}\)
- The domain of \((f + g)(x)\) is \((- \infty, \infty)\)