Exponential Functions

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	$e \approx 2.71828 \dots$

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 $b > 0$ , $b \neq 1$ :

Dom (f) = ℝ = (- \infty, \infty) and 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.

Graphs of exponential functions y = bˣ for bases b loading= — Graphs of exponential functions $y = b^{x}$ when $b > 1$ . All pass through $(0,1)$ and grow without bound as $x \to \infty$ .

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 $(0, \infty)$ .

Graphs of exponential functions y = bˣ for bases 0 < b < 1 (0.5ˣ, 0.2ˣ), showing exponential decay. — Graphs of exponential functions $y = b^{x}$ when $0 < b < 1$ . All pass through $(0,1)$ and decay toward zero as $x \to \infty$ .

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.

Graphs showing the symmetry between exponential functions with reciprocal bases (e.g., 2ˣ and (½)ˣ) across the y-axis. — The graphs of $y = b^{x}$ and $y = (1 / b)^{x} = b^{- x}$ are symmetric with respect to the $y$ -axis.

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

Comparison graph of exponential growth (2ˣ) versus polynomial growth (x²), showing the exponential eventually overtaking the polynomial. — The graphs of $y = 2^{x}$ and $y = x^{2}$ intersect at three points, but for $x > 4$ , the exponential $2^{x}$ dominates and the gap widens steeply.

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

(- \infty, \infty)

. To graph

f (x) = 1 - 3^{x}

, we use transformations starting from

y = 3^{x}

: Step 1. Begin with

y = 3^{x}

Graph of the exponential function y = 3ˣ. — Graph of $y = 3^{x}$

Step 2. Reflect in the

x

-axis to obtain

y = - 3^{x}

Graph showing the reflection of y = 3ˣ across the x-axis to form y = -3ˣ. — Graphs of $y = 3^{x}$ and $y = - 3^{x}$

Step 3. Shift upward 1 unit to obtain

y = 1 - 3^{x}

Graph showing the vertical shift of y = -3ˣ up 1 unit to form y = 1 - 3ˣ. — Graphs of $y = - 3^{x}$ and $y = 1 - 3^{x}$

The range of

f

(- \infty, 1)

. We verify algebraically: since

0 < 3^{x} < \infty

, we have

- \infty < - 3^{x} < 0 ⟹ - \infty < 1 - 3^{x} < 1.

(b) Again the domain is

ℝ

. To graph

g (x) = 3^{- x} - 2

: Step 1. Begin with

y = 3^{x}

(same starting graph). Step 2. Reflect in the

y

-axis to obtain

y = 3^{- x}

Graph showing the reflection of y = 3ˣ across the y-axis to form y = 3⁻ˣ. — Graphs of $y = 3^{x}$ and $y = 3^{- x}$

Step 3. Shift downward 2 units to obtain

y = 3^{- x} - 2

Graph showing the vertical shift of y = 3⁻ˣ down 2 units to form y = 3⁻ˣ - 2. — Graphs of $y = 3^{- x}$ and $y = 3^{- x} - 2$

The range of

g

(- 2, \infty)

. Verification: since

0 < 3^{x} < \infty

, a reflection in the

y

-axis does not change the range, so

0 < 3^{- x} < \infty

, and shifting down 2 gives

- 2 < 3^{- x} - 2 < \infty .

The Natural Exponential Function

The natural exponential function is $f (x) = e^{x}$ , where

e = 2.71828182845904523536 \dots

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 $π$ ), and its graph $y = e^{x}$ sits between $y = 2^{x}$ and $y = 3^{x}$ because $2 < e < 3$:

Graph of the natural exponential function y = eˣ plotted between y = 2ˣ and y = 3ˣ. — The natural exponential function $y = e^{x}$ , with $e = 2.71828 \dots$ , lies between $y = 2^{x}$ and $y = 3^{x}$ .

The natural exponential is called "the" exponential function because it is the unique function satisfying : 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

x \to - \infty

; when

0 < b < 1

it approaches

y = 0

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 $| a |$

Work through transformations step by step, tracking the

y

-intercept and asymptote at each stage.

Does the exponential function eventually beat every polynomial?

Yes. For any base

b > 1

and any polynomial

x^{n}

(no matter how large

n

is), the exponential

b^{x}

eventually dominates: as

x \to \infty

b^{x} / x^{n} \to \infty

. This is often stated as "exponentials grow faster than any polynomial." The crossover point may be very far to the right for small

b

or large

n

, but the exponential always wins in the long run.

What is the significance of

e

compared to other bases?

The number

e

is the unique base for which

\frac{d}{d x} [b^{x}] = b^{x}

exactly, with no extra constant factor. For any other base,

\frac{d}{d x} [b^{x}] = b^{x} \ln b

, which is only equal to

b^{x}

when

\ln b = 1

, i.e., when

b = e

. This self-referential derivative property makes

e^{x}

the natural choice for modeling continuous growth, solving differential equations, and simplifying integrals throughout calculus.