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−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 .

GPT-4o mini · Answer

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 $(- \infty, \infty)$.