Use the functions below.

Question

f(x)=3x+1
g(x)=|x+3|−5
What is (f⋅g)(x)
? What is the domain of (f⋅g)(x)
?

(1 point)
Responses

(f⋅g)(x)=3x|x+3|−5⋅3x+|x+3|−5
and the domain of (f⋅g)(x)
is (−∞, ∞)
.
open paren f times g close paren of x is equal to 3 to the x th power divides x plus 3 divides negative 5 times 3 to the x th power plus start absolute value x plus 3 end absolute value minus 5 and the domain of open paren f times g close paren of x is open paren negative infinity comma infinity close paren .

(f⋅g)(x)=3x|x+3|−5⋅3x+|x+3|−5
and the domain of (f⋅g)(x)
is x≥−3
.
open paren f times g close paren of x is equal to 3 to the x th power divides x plus 3 divides negative 5 times 3 to the x th power plus start absolute value x plus 3 end absolute value minus 5 and the domain of open paren f times g close paren of x is x is greater than or equal to negative 3 .

(f⋅g)(x)=3x+|x+3|−4
and the domain of (f⋅g)(x)
is x≥−3
.
open paren f times g close paren of x is equal to 3 to the x th power plus start absolute value x plus 3 end absolute value minus 4 and the domain of open paren f times g close paren of x is x is greater than or equal to negative 3 .

(f⋅g)(x)=3x+|x+3|−4
and the domain of (f⋅g)(x)
is (−∞, ∞)

Answer 1

To find \((f \cdot g)(x)\), we need to multiply the functions \(f(x)\) and \(g(x)\) together.

1. The functions are:

   • \(f(x) = 3x + 1\)
   • \(g(x) = |x + 3| - 5\)

2. Thus, \((f \cdot g)(x) = f(x) \cdot g(x) = (3x + 1)(|x + 3| - 5)\).

Now we will expand this expression:

\[ (f \cdot g)(x) = (3x + 1)(|x + 3| - 5) \]

Distributing:

\[ = (3x + 1)|x + 3| - 5(3x + 1) \]

This expression cannot be further simplified without specific values of \(x\) since \( |x + 3| \) will change its form based on the value of \(x\). However, we can keep it in this factored form.

Now, let's look at the domain of \((f \cdot g)(x)\).

1. The domain of \(f(x) = 3x + 1\) is all real numbers \((-\infty, \infty)\) since it is a linear function.
2. The domain of \(g(x) = |x + 3| - 5\) is also all real numbers since the absolute value function can take any real number as input.

Therefore, since both of the component functions have a domain of \((-\infty, \infty)\), the domain of \((f \cdot g)(x)\) is also:

\[ (-\infty, \infty) \]

In conclusion:

\[ (f \cdot g)(x) = (3x + 1)(|x + 3| - 5) \]

and the domain of \((f \cdot g)(x)\) is:

\[ (-\infty, \infty) \]