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−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 first need to define the functions clearly:

1. \( f(x) = -2x^2 + 4x - 7 \)

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

Now, let's calculate \((f + g)(x)\):

\[
(f + g)(x) = f(x) + g(x)
\]

Substituting \(f(x)\) and \(g(x)\):

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

Simplifying this expression:

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

Combining like terms:

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

So, we have:

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

Next, we need to determine the domain of \((f + g)(x)\).

1. The function \(f(x)\) is a polynomial, which means it is defined for all real numbers \((-\infty, \infty)\).

2. The function \(g(x) = -\sqrt[3]{x - 1} + 2\) is also defined for all real numbers because cube roots are defined for all real values.

Since both functions are defined for all real numbers, the combined domain for \((f + g)(x)\) is also \((-\infty, \infty)\).

Thus, the final results are:

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

The domain of \((f + g)(x)\) is \((- \infty, \infty)\).