Use the functions below.

Question

ONLY USE ANSWERS IVE GIVEN YOU.
f(x)=−x4+3x2−x+1
g(x)=−3x+1−−−−−√−5
What is (f−g)(x)
? What is the domain of (f−g)(x)
?

(1 point)
Responses

(f−g)(x)=−x4+3x2−x+3x+1−−−−−√+6
and the domain of (f−g)(x)
is (−∞,∞)
.
open paren f minus g close paren of x is equal to negative x to the 4th power plus 3 x squared minus x plus square root of 3 x plus 1 end root plus 6 and the domain of open paren f minus g close paren of x is open paren negative infinity comma infinity close paren .

(f−g)(x)=−x4+3x2−x+3x+1−−−−−√+6
and the domain of (f−g)(x)
is x≥−13
.
open paren f minus g close paren of x is equal to negative x to the 4th power plus 3 x squared minus x plus square root of 3 x plus 1 end root plus 6 and the domain of open paren f minus g close paren of x is x is greater than or equal to negative 1 third .

(f−g)(x)=−x4+3x2−x+3x+1−−−−−√−4
and the domain of (f−g)(x)
is x≥−13
.
open paren f minus g close paren of x is equal to negative x to the 4th power plus 3 x squared minus x plus square root of 3 x plus 1 end root minus 4 and the domain of open paren f minus g close paren of x is x is greater than or equal to negative 1 third .

(f−g)(x)=−x4+3x2−x+3x+1−−−−−√−4
and the domain of (f−g)(x)
is (−∞,∞)
.
open paren f minus g close paren of x is equal to negative x to the 4th power plus 3 x squared minus x plus square root of 3 x plus 1 end root minus 4 and the domain of open paren f minus g close paren of x is open paren negative infinity comma infinity close paren .
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Answer 1

To find \( (f-g)(x) \), we start with the definitions of \( f(x) \) and \( g(x) \):

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

Now we can simply calculate \( (f-g)(x) \):

\[ (f-g)(x) = f(x) - g(x) = -x^4 + 3x^2 - x + 1 - \left( \sqrt{-3x + 1} - 5 \right) \] \[ = -x^4 + 3x^2 - x + 1 - \sqrt{-3x + 1} + 5 \] \[ = -x^4 + 3x^2 - x + 6 - \sqrt{-3x + 1} \]

Thus,

\[ (f-g)(x) = -x^4 + 3x^2 - x + 6 - \sqrt{-3x + 1} \]

Next, we need to consider the domain of \( (f-g)(x) \). The term \( \sqrt{-3x + 1} \) requires its radicand to be non-negative:

\[ -3x + 1 \geq 0 \] \[ 1 \geq 3x \] \[ \frac{1}{3} \geq x \]

or

\[ x \leq \frac{1}{3} \]

So the domain of \( (f-g)(x) \) is \( (-\infty, \frac{1}{3}] \).

Based on this, the correct response is:

\[ (f-g)(x) = -x^4 + 3x^2 - x + 6 - \sqrt{-3x + 1}, \quad \text{and the domain of } (f-g)(x) \text{ is } (-\infty, \frac{1}{3}]. \]