Question

Priya is comparing two functions, f(x) = 10x^2 and g(x) = 3(2)^x, to find out which one grows at a faster rate as its x values get very large. She notices that f(1) = 10 and f(2) = 40. However, g(1) = 6 and g(2) = 12. She concludes that as x continues to grow, the values of f will be greater than the values of g. Explain or show why Priya's conclusion is incorrect.

A. Priya's conclusion is correct because f grows faster than g.
B. Priya's conclusion is incorrect because g grows faster than f.
C. Priya's conclusion is correct because f and g have the same growth rate.
D. Priya's conclusion is incorrect, and further analysis is needed to determine which function grows faster.

Answer 1

Priya's conclusion that the function f(x) grows faster than g(x) is incorrect because g(x) is exponential and will eventually outpace the quadratic growth of f(x) as x becomes very large.

Step-by-step explanation:

Priya is trying to compare the growth rates of two functions, f(x) = 10x^2 and g(x) = 3(2)^x, as x increases. Initially, Priya concludes that since f(1) = 10 and f(2) = 40 are both larger than g(1) = 6 and g(2) = 12, the function f(x) grows faster than g(x). However, this conclusion is incorrect because the function g(x) is exponential, and exponential growth rates eventually outpace polynomial growth rates such as that of f(x), so choice B is correct, meaning Priya's conclusion is incorrect because g grows faster than f.