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 (−∞,∞).  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 (−∞,∞).  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 open paren negative infinity comma infinity close paren .(f⋅g)(x)=9x+7⋅3x+10 and the domain is x≥0.

Answer 1

To determine how many bacteria are in both petri dishes after \( x \) days, we need to consider both functions \( f(x) \) and \( g(x) \).

1. **Finding \( f(x) + g(x) \)**:
\[
f(x) = 3x + 2
\]
\[
g(x) = 3x + 5
\]

Adding both functions:
\[
f(x) + g(x) = (3x + 2) + (3x + 5) = 6x + 7
\]

2. **Finding the Domain**:
Since \( x \) represents the number of days, it should be non-negative. Thus, the domain for the combined function \( f + g \) is:
\[
x \geq 0
\]

So the answer is:
- \( (f + g)(x) = 6x + 7 \) and the domain is \( x \geq 0 \).

Now let's analyze the options you provided:

1. **(f⋅g)(x)=9x+7⋅3x+10 and the domain is (−∞,∞)**: This is incorrect since it represents multiplication, not addition.
2. **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 greater than or equal to 0**: Incorrect because \( 2 \times 3^x + 7 \) does not match our addition.
3. **(f+g)(x)=2⋅3x+7 and the domain is (−∞,∞)**: Incorrect because the expression does not match our sum and the domain should not be \( (-\infty, \infty) \).
4. **(f⋅g)(x)=9x+7⋅3x+10 and the domain is x≥0**: Incorrect as it describes multiplication again.

Therefore, the correct expression is:
- \( (f + g)(x) = 6x + 7 \) and the domain is \( x \geq 0 \).