To find the function that represents the average number of bacteria in the culture after \( x \) days, we need to divide the total number of bacteria \( f(x) \) over the time period. The average number of bacteria over the interval from day 0 to day \( x \) can be represented as follows:
\[ \text{Average Number of Bacteria} = \frac{1}{x} \int_0^x f(t) , dt \]
where \( f(t) = t^4 - 3t^2 \).
So, we first need to compute the integral \( \int_0^x (t^4 - 3t^2) , dt \):
- Compute the integral:
\[ \int (t^4 - 3t^2) , dt = \frac{t^5}{5} - t^3 + C \]
- Now evaluate the definite integral from 0 to \( x \):
\[ \int_0^x (t^4 - 3t^2) , dt = \left[ \frac{t^5}{5} - t^3 \right]_0^x = \left( \frac{x^5}{5} - x^3 \right) - \left( 0 - 0 \right) = \frac{x^5}{5} - x^3 \]
- Now plug this into the average formula:
\[ \text{Average Number of Bacteria} = \frac{1}{x} \left( \frac{x^5}{5} - x^3 \right) = \frac{x^5}{5x} - \frac{x^3}{x} = \frac{x^4}{5} - x^2 \]
Therefore, the average number of bacteria after \( x \) days is represented by the function:
\[ A(x) = \frac{x^4}{5} - x^2 \]
Thus, the final answer for \( (fg)(x) \) which typically denotes the average in this context is:
\[ (fg)(x) = \frac{x^4}{5} - x^2 \]
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