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|−4
 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 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 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|−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 .

GPT-4o mini · Answer

To find $(f \cdot g)(x)$, we need to multiply $f(x)$ and $g(x)$. Given: $$ f(x) = 3x + 1 $$ $$ g(x) = |x + 3| - 5 $$ The product $(f \cdot g)(x)$ can be calculated as follows: $$ (f \cdot g)(x) = f(x) \cdot g(x) = (3x + 1) \cdot (|x + 3| - 5) $$ Now let’s expand this: $$ (f \cdot g)(x) = (3x + 1)(|x + 3| - 5) = (3x + 1)|x + 3| - 5(3x + 1) $$ $$ = (3x + 1)|x + 3| - (15x + 5) = (3x + 1)|x + 3| - 15x - 5 $$ This means: $$ (f \cdot g)(x) = (3x + 1)|x + 3| - 15x - 5 $$ Now simplify: $$ = 3x|x + 3| + |x + 3| - 15x - 5 $$ The domain of $(f \cdot g)(x)$ is determined by the domains of $f(x)$ and $g(x)$. Both functions $f(x)$ and $g(x)$ are defined for all real numbers. The domain of $(f \cdot g)(x)$ is therefore: $$ (-\infty, \infty) $$ ### Conclusion: $(f \cdot g)(x) = (3x + 1)|x + 3| - 15x - 5$ and the domain of $(f \cdot g)(x)$ is $(-∞, ∞)$. However, it looks like none of the provided response options precisely reflect this outcome. The closest conclusion would be that the domain is indeed $(-∞, ∞)$. Please ensure to check the representation in the options you provided to see if similar structures are mentioned so you can select the most appropriate response based on my detailed explanation.