Exponential functions

With a > 0, the base decides everything. If b > 1, the function is increasing everywhere: each step multiplies by more than 1. If 0 < b < 1, it is decreasing everywhere: each step multiplies by less than 1. There is no third case worth memorizing, because b = 1 is constant and the definition excludes it.

Either way the graph has no maximum, no minimum, and no inflection point. It never turns around: an exponential that rises, rises forever; one that falls, falls forever, always by the same factor per step. If a sketch of an exponential has a hump or a flat spot in the middle, the sketch is wrong before you check a single value.

The single most damaging arithmetic slip in this unit: reading $b^{-n}$ as a negative number. It is a reciprocal: $b^{-n} = 1/b^n$. So $2^{-3} = 1/8$, a small positive number, and $2^x$ at x = −10 is $1/1024$, tiny but still positive.

Consequences worth spelling out:

1. The outputs of $a \cdot b^x$ with a > 0 are positive for every real x. Negative inputs shrink the output toward 0; they never push it below 0.
2. The output never reaches 0 either. $1/b^n$ gets as small as you like but is never zero, so the range is $(0, \infty)$.
3. $b^0 = 1$, not 0. That is why f(0) = a: the base contributes a factor of 1 at x = 0.

Because the outputs approach 0 without reaching it, every $f(x) = a \cdot b^x$ has a horizontal asymptote at y = 0. For a growth function (b > 1) the approach happens as $x \to -\infty$; for a decay function (0 < b < 1) it happens as $x \to +\infty$. In limit language for growth: $\lim_{x \to -\infty} a b^x = 0$ and $\lim_{x \to +\infty} a b^x = \infty$.

The asymptote is a statement about ends, and the graph hugs it from one side only. Meanwhile the domain is all real numbers: an exponential function accepts any input, including negative numbers and fractions. It is the sequence 2, 4, 8, 16 with every real-number gap filled in, not a list that starts at zero.

Real problems speak in percents, but the function runs on factors: b = 1 + r, with r the per-step percent change as a decimal. Growth of 3% per year is b = 1.03. Decay of 15% per step is b = 1 − 0.15 = 0.85: the factor is the fraction kept. The base 0.85 does not mean an 85% loss, and a base of 0.03 would mean losing 97% per step.

One more habit: when outputs shrink by halves, the factor is 1/2 and k steps compound to $(1/2)^k$. Four halvings leave $1/16$, not $1/8$ and not $4 \times 1/2 = 2$. Count the steps, then compound the factor that many times.