Mathematics

Problem

Consider the two parent functions

f(x)=x^{2}

and

g(x)=2^{x}

. a. (i) Calculate the values of

f(5)

and

g(5)

. b. (ii) Using the results from part (i), determine which function grows faster for large positive values of

x

Intuitive explanation

The key insight is recognizing how these two fundamentally different function families—polynomial and exponential—grow at different rates. Start by comparing their behavior: quadratic growth is steady and predictable, while exponential growth is slow at first but eventually dominates everything. Plotting or evaluating both at several points (negative, zero, positive) will reveal where they intersect and which one "wins" in different regions, helping you understand why exponential functions are so powerful in real-world modeling.

Intuitive explanation

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