Exponential Functions

Imagine you have a snowball rolling downhill — the bigger it gets, the faster it grows, because more surface area picks up more snow. That's exponential growth in a nutshell. In math, we write it as $f(x) = b^x$, where the variable is in the exponent instead of the base. This one change makes the function grow (or shrink) incredibly fast compared to anything you've seen before.

The number **$e ≈ 2.718$** is a special constant that shows up whenever growth is continuous — like bacteria dividing nonstop or interest compounding every instant. Think of $e$ like $\pi$: it's a fixed number with a special meaning. When a bank says "compounded continuously," they use $A = Pe^{rt}$. When they say monthly or quarterly, they use $A = P(1 + r/n)^{nt}$ instead.

Compound interest is the real-world superpower of exponential functions. You deposit some money ($P$), the bank pays you interest, and then you earn interest on *that* interest too. The more often they compound (monthly vs. yearly), the more your money grows — and continuous compounding ($Pe^{rt}$) is the ultimate limit of that process.

The biggest trap on exam problems: forgetting to turn the interest rate into a decimal. If the problem says 6%, you must plug in $r = 0.06$, not 6. One misplaced decimal point and your answer will be wildly off. Also remember: if $b > 1$ the function grows, and if $0 < b < 1$ it decays (shrinks toward zero).