Exercises: Compare Exponential and Polynomial Growth

$g(x) = 2^x$, because exponential functions are always larger.

$f(x) = 100x$, because the linear function is larger at $x = 5$ ($f(5) = 500$ vs $g(5) = 32$).

They are equal at every point.

Neither — both are undefined at $x = 5$.

Yes — $x^2$ accelerates (increases at an increasing rate), so it must be exponential.

No — exponential growth requires CONSTANT RATIOS between consecutive outputs. The ratios of $x^2$ are not constant.

Yes — any function that grows faster and faster is exponential.

No — $x^2$ does not get faster and faster; it grows at a steady linear rate.

The linear function will always be larger because it has a bigger coefficient (500 vs 1.5).

The crossover occurs between $x = 20$ and $x = 25$, after which the exponential dominates.

The exponential is always larger because $1.5^{25} > 500 \times 25$.

Neither function dominates — they oscillate in which is larger.

No — the base 1.01 is too close to 1 for the exponential to ever catch up.

No — $x^{100}$ has such a high degree that no exponential can exceed it.

Yes — for any base $b > 1$, the exponential $b^x$ eventually exceeds every polynomial, regardless of the polynomial's degree or coefficients.

Only if the base is at least 2.

$f(x) > g(x)$ for all $x > 0$.

$g(x) > f(x)$ for all $x > 0$.

$f(x) > g(x)$ for small $x$ values (around $x = 5$, say), but $g(x) > f(x)$ for all sufficiently large $x$.

$g(x) > f(x)$ for all $x$ between 0 and 5, then $f(x) > g(x)$ afterwards.