Discuss in easy words and in a paragraph the criteria for arranging a set of exponential and quadratic growth functions in order from fastest growth to slowest as x increases toward infinity.(2 points)

Answer 1

When we want to compare exponential and quadratic growth functions to see which grows faster as x gets larger, we look at how they behave as x approaches infinity. Exponential functions (like \(e^x\) or \(2^x\)) start off slow but grow much faster than quadratic functions (like \(x^2\) or \(x^3\)) when x becomes very large. This is because exponential functions increase at a rate that keeps getting bigger, while quadratic functions grow at a steady rate that doesn’t increase as quickly. So, when we arrange these functions from fastest to slowest growth as x increases, the order would be: exponential functions grow the fastest, followed by polynomial functions (like quadratic ones), and then linear functions (like \(x\)), which grow even slower. At infinity, exponential growth always outpaces quadratic growth.