Question

Use the functions below.

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−−−−−√−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 .

(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 .
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Answer 1

To find \((f - g)(x)\), we need to subtract \(g(x)\) from \(f(x)\).

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

First, let's clarify what \((f - g)(x)\) means:
\[
(f - g)(x) = f(x) - g(x) = f(x) - \left( \sqrt{-3x + 1} - 5 \right)
\]
This simplifies to:
\[
(f - g)(x) = -x^4 + 3x^2 - x + 1 + 5 - \sqrt{-3x + 1}
\]
\[
= -x^4 + 3x^2 - x + 6 - \sqrt{-3x + 1}
\]

Next, let's look at the domain of \((f - g)(x)\). The only restriction comes from the square root function in \(g(x)\). We need to ensure that the expression inside the square root is non-negative:
\[
-3x + 1 \geq 0
\]
Solving this inequality:
\[
-3x \geq -1
\]
\[
x \leq \frac{1}{3}
\]

So, the domain for \((f - g)(x)\) is:
\[
(-\infty, \frac{1}{3}]
\]

Putting it all together:
\[
(f - g)(x) = -x^4 + 3x^2 - x + 6 - \sqrt{-3x + 1}
\]
The domain is \(x \leq \frac{1}{3}\).

From the provided options, here’s the correct interpretation:
\[
(f−g)(x) = -x^4 + 3x^2 - x + 6 - \sqrt{-3x + 1}
\]
and the domain of \((f−g)(x)\) is \(x \leq \frac{1}{3}\).

None of the options you provided seem to match this exactly, but the closest one is the second option:
\[
(f−g)(x) = -x^4 + 3x^2 − x + \sqrt{3x + 1} − 4 \text{ with domain } x \ge -\frac{1}{3}
\]

So, we have:
1. **Expression for \((f-g)(x)\)**: \( -x^4 + 3x^2 - x + 6 - \sqrt{-3x + 1} \)
2. **Domain of \((f - g)(x)\)**: \((-∞, \frac{1}{3}]\)