Topic 2.3 Exponential Functions: define exponential functions, describe how the base and initial value determine growth or decay, and analyze the domain, range and horizontal asymptote of the graph.

What this topic is asking

The College Board (Topic 2.3) wants you to define an exponential function and analyze its graph. You must identify the initial value and the base, decide whether the function shows growth or decay, and state the domain, range and horizontal asymptote. You should also recognize the natural base $e$.

The form of an exponential function

The base $b = 1$ is excluded because $1^x$ is the constant function, and negative bases are excluded because they fail to give real outputs for many inputs.

Growth versus decay

The base alone decides growth or decay; the initial value $a$ only scales the graph vertically and fixes the $y$-intercept.

Domain, range and the asymptote

An exponential function $a \cdot b^x$ accepts every real input, so its domain is all real numbers. With $a > 0$ the outputs are always positive, so the range is $y > 0$, and the graph approaches but never reaches the horizontal asymptote $y = 0$. A vertical shift $a \cdot b^x + k$ moves the asymptote to $y = k$ and the range to $y > k$ (or $y < k$ if $a < 0$).

The natural base e

The base $e \approx 2.71828$ is the natural base, and $f(x) = e^x$ is the natural exponential function. It models continuous growth (compound interest taken to its limit, continuous population growth, radioactive decay written as $e^{kt}$). Because $e > 1$, $e^x$ grows; because $0 < e^{-1} < 1$, $e^{-x}$ decays. The natural exponential is the function whose inverse is the natural logarithm $\ln$, which is why it threads through the rest of the unit.

A point worth making once is that an exponential function never reaches its horizontal asymptote, no matter how far out you go. The outputs get arbitrarily close to $y = 0$ but stay strictly positive, because a positive base raised to any power is positive. This is the graphical meaning of the range being $y > 0$ rather than $y \geq 0$, and it is what distinguishes the unbounded-then-flattening shape of an exponential from a polynomial. Recognizing that the asymptote is approached but never touched also explains why exponential decay models, such as drug concentration or cooling, predict a quantity that nears but never equals zero.

Try this

Q1. Does $f(x) = 2 \cdot 1.5^{x}$ grow or decay, and what is its $y$-intercept? [1 point]

• Cue. Base $1.5 > 1$, so it grows; the $y$-intercept is $f(0) = 2$.

Q2. What is the range of $g(x) = 3^x + 4$? [1 point]

• Cue. The vertical shift raises the asymptote to $y = 4$, so the range is $y > 4$.