Use the functions below.

Question

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 functions:

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

Next, we need to compute \((f - g)(x)\):

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

This simplifies to:

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

Substituting for \(f(x)\):

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

So, we have:

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

Now, let's determine the domain of \((f - g)(x)\). The domain is determined by the expression inside the square root of \(g(x)\), since the square root function requires that its argument be non-negative:

\[ -3x + 1 \geq 0 \]

Solving:

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

Thus, the domain is

\[ (-\infty, \frac{1}{3}] \]

Putting this all together, the final answers are:

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

\[ (-\infty, \frac{1}{3}] \]