Question

The number of bacteria in a petri dish after x days is represented by the function f(x)=3x+2
. The number of bacteria in another petri dish after x days is represented by the function g(x)=3x+5
. How many bacteria are in both petri dishes after x days? What is the domain?(1 point)
Responses

(f⋅g)(x)=9x+7⋅3x+10
and the domain is x≥0
.
open paren f times g close paren of x is equal to 9 to the x th power plus 7 times 3 to the x th power plus 10 and the domain is x is greater than or equal to 0 .

(f⋅g)(x)=9x+7⋅3x+10
and the domain is (−∞,∞)
.
open paren f times g close paren of x is equal to 9 to the x th power plus 7 times 3 to the x th power plus 10 and the domain is open paren negative infinity comma infinity close paren .

(f+g)(x)=2⋅3x+7
and the domain is x≥0
.
open paren f plus g close paren of x is equal to 2 times 3 to the x th power plus 7 and the domain is x is greater than or equal to 0 .

(f+g)(x)=2⋅3x+7
and the domain is (−∞,∞)
.

Answer 1

To find the number of bacteria in both petri dishes after \(x\) days, we can look at both the product and the sum of the functions \(f(x)\) and \(g(x)\).

1. **Sum of the functions**:
\[
(f+g)(x) = f(x) + g(x) = (3x + 2) + (3x + 5) = 6x + 7
\]

2. **Product of the functions**:
\[
(f \cdot g)(x) = f(x) \cdot g(x) = (3x + 2) \cdot (3x + 5)
\]
To find this, we can use the distributive property:
\[
(3x + 2)(3x + 5) = 9x^2 + 15x + 6x + 10 = 9x^2 + 21x + 10
\]

Now let's summarize the results:

- The **sum** of the bacteria in the petri dishes after \(x\) days is:
\[
(f+g)(x) = 6x + 7
\]

- The **product** of the bacteria in the petri dishes after \(x\) days is:
\[
(f \cdot g)(x) = 9x^2 + 21x + 10
\]

### Domain:
Since both functions represent the number of bacteria, the domain for \(x\) (the number of days) is non-negative:
\[
\text{Domain: } x \geq 0
\]

Thus, the answers for the functions and the domain are:

- \( (f+g)(x) = 6x + 7 \) and the domain is \( x \geq 0 \).
- \( (f \cdot g)(x) = 9x^2 + 21x + 10 \) and the domain is also \( x \geq 0 \).

Final answer:
1. **Sum**: \( (f+g)(x) = 6x + 7 \) and domain is \( x \geq 0 \).
2. **Product**: \( (f \cdot g)(x) = 9x^2 + 21x + 10 \) and domain is \( x \geq 0 \).