To determine which function corresponds to the graph of the population growth of a swarm of mosquitoes, we should note that population growth can typically be modeled using exponential functions of the form:
\[ f(x) = a \cdot b^x \]
where:
- \( a \) represents the initial population,
- \( b \) is the growth factor (if \( b > 1 \), the population is growing; if \( 0 < b < 1 \), the population is decreasing).
From the given options:
- \( f(x) = 5(1.3)^x \) - This indicates growth since \( b = 1.3 > 1 \).
- \( f(x) = 1.3(5)^x \) - This also indicates exponential growth, but the starting point would be at 1.3 and grow rapidly due to the base being 5.
- \( f(x) = 5(0.3)^x \) - This indicates decay since \( b = 0.3 < 1 \).
- \( f(x) = 5(0.7)^x \) - This also indicates decay since \( b = 0.7 < 1 \).
Based on what we typically expect from mosquito population growth (which is generally exponential growth in an uncontrolled environment), the most appropriate function from this list is:
\( f(x) = 5(1.3)^x \)
This corresponds to a scenario where the mosquito population starts at 5 and grows over time.