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−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−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 . (f+g)(x)=−2x2+4x−x−6−−−−−√3 and the domain of (f+g)(x) is x≥1.

Answer 1

To find \((f+g)(x)\), we first need to separately write down the functions \(f(x)\) and \(g(x)\).

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

Now, we add these two functions together:

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

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

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

Next, we consider the domain of \((f+g)(x)\):
- The function \(-2x^2 + 4x - 5\) is a polynomial and defined for all real numbers.
- The term \(-\sqrt[3]{x - 1}\) is also defined for all real numbers, as cube root functions are defined for any real number.

Thus, the overall function \((f+g)(x)\) is defined for all real numbers.

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

Based on the options provided, the correct choice seems to be:
\[
(f+g)(x)=-2x^2+4x-\sqrt[3]{x-1}-5 \quad \text{and the domain of } (f+g)(x) \text{ is } (-\infty, \infty).
\]