comparing rates of change\nwhich function increases at a faster rate on 0 to infinity, f(x) = x² or g(x) =…

Explanation:

Step1: Recall growth - rate concept

Exponential functions grow faster than polynomial functions in the long - run.

Step2: Analyze function types

$f(x)=x^{2}$ is a polynomial function of degree 2. $g(x) = 2^{x}$ is an exponential function with base 2.

Step3: Consider limit

We can consider the limit $\lim_{x\rightarrow\infty}\frac{2^{x}}{x^{2}}$. Using L'Hopital's rule (differentiating numerator and denominator successively), $\lim_{x\rightarrow\infty}\frac{2^{x}\ln 2}{2x}$, and applying L'Hopital's rule again $\lim_{x\rightarrow\infty}\frac{2^{x}(\ln 2)^{2}}{2}=\infty$. This shows that $2^{x}$ grows faster than $x^{2}$ as $x\rightarrow\infty$.

Answer:

$g(x) = 2^{x}$ increases faster because exponential functions grow faster than polynomial functions as $x$ approaches infinity.