An exponential function $f(x) = b^x$ places the variable in the exponent rather than the base. This simple shift produces functions with remarkable properties: they grow (or decay) at a rate proportional to their current value, they are defined for all real $x$, and they are always positive. Understanding their graphs and transformations is the foundation for logarithms, compound interest, population models, and calculus.
Quick Reference
| Property | Formula / Value |
|---|---|
| General form | $f(x) = b^x$, where $b > 0$, $b \neq 1$ |
| Domain | $(-\infty, \infty)$ |
| Range | $(0, \infty)$ |
| $y$-intercept | $(0, 1)$ for every base $b$ |
| Asymptote | $x$-axis ($y = 0$) |
| $b > 1$ | Increasing (exponential growth) |
| $0 < b < 1$ | Decreasing (exponential decay) |
| Reflection in $y$-axis | $(1/b)^x = b^{-x}$ |
| Natural base | $e \approx 2.71828\ldots$ |
Definition and Basic Properties
A function of the form
$f(x) = b^x$where $b > 0$ is a fixed constant and $x$ is the variable is called an exponential function with base $b$. The variable appears in the exponent, which is what makes it "exponential."
The requirement $b > 0$ is essential: when $b < 0$, the expression $b^x$ is only defined for specific rational values of $x$ (those with odd-integer denominators), so the function would not have a continuous domain. Requiring $b > 0$ ensures $f(x) = b^x$ is defined for all real $x$. The case $b = 1$ gives the constant function $1^x = 1$, which is uninteresting.
Three key properties hold for every exponential function $f(x) = b^x$ (with $b > 0$, $b \neq 1$):
- The expression $b^x$ is defined for all real $x$, so the domain of $f$ is $(-\infty, \infty)$.
- Since $b^x > 0$ for any $x$, the graph of $f$ always lies above the $x$-axis; the range is $(0, \infty)$.
- Since $b^0 = 1$, the graph passes through $(0, 1)$: every exponential function shares the same $y$-intercept.
If $f(x) = b^x$ with $b > 0$, $b \neq 1$:
$\operatorname{Dom}(f) = \mathbb{R} = (-\infty,\infty) \qquad \text{and} \qquad \operatorname{Rng}(f) = (0,\infty).$
Graphs of Exponential Functions
The shape of the graph depends critically on whether $b > 1$ or $0 < b < 1$.
When b > 1: Exponential Growth
When $b > 1$, the function $y = b^x$ increases as $x$ increases. For large positive $x$, the value of $y$ becomes very large. For large negative $x$, the value of $y$ approaches zero from above: the graph hugs the negative $x$-axis but never touches it. The $x$-axis is a horizontal asymptote.
When $0 < b < 1$: Exponential Decay
When $0 < b < 1$, the function $y = b^x$ decreases as $x$ increases. As $x \to +\infty$, the value approaches zero; as $x \to -\infty$, the value grows without bound. The range is still $(0, \infty)$.
Symmetry Between Reciprocal Bases
The graphs of $y = b^x$ and $y = (1/b)^x$ are reflections of each other in the $y$-axis. To see why, note that
$(1/b)^x = b^{-x}.$The function $y = b^{-x}$ takes the same values for positive $x$ as $y = b^x$ takes for negative $x$ of the same magnitude, and vice versa. This is exactly what reflection in the $y$-axis does.
Because $1/b > 1$ whenever $0 < b < 1$, and $0 < 1/b < 1$ whenever $b > 1$, every decaying exponential corresponds to a growing one obtained by replacing $b$ with $1/b$.
Exponential Growth Vs. Polynomial Growth
When $b > 1$, the exponential function $y = b^x$ eventually outgrows any polynomial $y = x^n$, no matter how large $n$ is. For example, with $b = 1.1$ and $n = 10$: at $x = 1000$, the polynomial $x^{10}$ equals $10^{30}$, while $1.1^{1000} \approx 2.47 \times 10^{41}$, roughly $2.5 \times 10^{11}$ times larger.
Graphing Transformations of Exponential Functions
Sketch the graph of each function and determine its domain and range.
- $f(x) = 1 - 3^x$
- $g(x) = 3^{-x} - 2$
Solution
(a) Since the base 3 is positive, there is no restriction on $x$. The domain of $f$ is $(-\infty, \infty)$. To graph $f(x) = 1 - 3^x$, we use transformations starting from $y = 3^x$: Step 1. Begin with $y = 3^x$:
The Natural Exponential Function
The natural exponential function is $f(x) = e^x$, where
$e = 2.71828182845904523536\ldots$is an irrational constant called Euler's number (also known as Napier's constant). The function $e^x$ is also written $\exp(x)$.
The number $e$ is irrational (like $\pi$), and its graph $y = e^x$ sits between $y = 2^x$ and $y = 3^x$ because $2 < e < 3$:
The natural exponential is called "the" exponential function because it is the unique function satisfying $f'(x) = f(x)$: its rate of change equals its current value at every point. Many calculators have a dedicated key for $e^x$, and note that $e = \exp(1)$.
Frequently Asked Questions
Why can't the base of an exponential function be negative?
When the base $b$ is negative, $b^x$ is only defined when $x$ is a rational number $p/q$ with $q$ odd. This leaves the function undefined for irrational $x$ and many rationals, so it cannot be graphed as a continuous curve. Calculus requires functions to be defined for all real $x$, which is why exponential functions require $b > 0$.What is the horizontal asymptote of an exponential function?
The horizontal asymptote of $y = b^x$ is the $x$-axis, i.e., $y = 0$. When $b > 1$ the curve approaches $y = 0$ as $x \to -\infty$; when $0 < b < 1$ it approaches $y = 0$ as $x \to +\infty$. Transformations shift this asymptote: for example, $y = b^x + k$ has horizontal asymptote $y = k$.How do I graph transformations of exponential functions?
Use the standard transformation toolkit applied to $y = b^x$:- $y = b^{x-h}$: horizontal shift right by $h$
- $y = b^x + k$: vertical shift up by $k$
- $y = -b^x$: reflection in the $x$-axis (range becomes $(-\infty, 0)$)
- $y = b^{-x}$: reflection in the $y$-axis (equivalent to $(1/b)^x$)
- $y = a \cdot b^x$: vertical stretch by factor $|a|$
