Use the functions below.

Question

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 \((- \infty, \infty)\)