The number of bacteria in a petri dish after x days is represented by the function f(x)=3x+2

Question

. 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 \( x \geq 0 \).