Use the functions below.

Question

f(x)=x−1−−−−√
g(x)=3x−1
What is(fg)(x)
? What is the domain of (fg)(x)
?

(1 point)
Responses

(fg)(x)=x√3
and the domain of (fg)(x)
is x≥1
.
open paren f over g close paren of x is equal to the fraction with numerator square root of x and denominator 3 and the domain of open paren f over g close paren times x times is x is greater than or equal to 1 .

(fg)(x)=x−1√3x−1
and the domain of (fg)(x)
is (−∞, ∞)
.
open paren f over g close paren of x is equal to the fraction with numerator square root of x minus 1 end root and denominator 3 x minus 1 and the domain of open paren f over g close paren of x is open paren negative infinity comma infinity close paren .

(fg)(x)=x−1√3x−1
and the domain of (fg)(x)
is x≥1
.
open paren f over g close paren of x is equal to the fraction with numerator square root of x minus 1 end root and denominator 3 x minus 1 and the domain of open paren f over g close paren times x times is x is greater than or equal to 1 .

(fg)(x)=x√3
and the domain of (fg)(x)
is (−∞, ∞)
.
open paren f over g close paren of x is equal to the fraction with numerator square root of x and denominator 3 and the domain of open paren f over g close paren of x is open paren negative infinity comma infinity close paren .

Answer 1

To find \((fg)(x)\), we need to multiply the two functions \(f(x)\) and \(g(x)\):

1. Given Functions:

   • \(f(x) = \sqrt{x - 1}\)
   • \(g(x) = 3x - 1\)

2. Calculating \((fg)(x)\): \[ (fg)(x) = f(g(x)) = f(3x - 1) \] Substituting \(g(x)\) into \(f\): \[ f(3x - 1) = \sqrt{(3x - 1) - 1} = \sqrt{3x - 2} \]

   Therefore, \((fg)(x) = \sqrt{3x - 2}\).

3. Finding the Domain of \((fg)(x)\): The expression under the square root must be non-negative: \[ 3x - 2 \geq 0 \] Solving this inequality: \[ 3x \geq 2 \quad \Rightarrow \quad x \geq \frac{2}{3} \]

Thus, the domain of \((fg)(x)\) is \(x \geq \frac{2}{3}\).

So, the correct response is: \[ (fg)(x) = \sqrt{3x - 2} \] and the domain of \((fg)(x)\) is \(x \geq \frac{2}{3}\).

Since none of the options provided correspond to this analysis directly:

• If \(\sqrt{3x - 2}\) is included in the choices, that is the right function.
• The domain being constraints related to \((-\infty, \infty)\) or other values does not apply according to our findings.

If you have any further options or need clarification, please let me know!