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−−−−−√−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 .
Skip to navigation

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 \((f - g)(x)\): \((-∞, \frac{1}{3}]\)