Exponential

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Our core exponential functions look like $r^x$. Not every exponential function can be represented with such formulas.

If we allow a leading coefficient, then we get a template that covers all the exponential functions. We can accomplish this through composition with linear functions.

Once we are familiar with this core template, we can describe all exponential functions through composition with linear functions.

Core Exponential Functions

\[ exp(x) = r^x \, \text { with } \, \text { and } \, 0 > r > 1 \]

The behavior of the core exponential functions is completely dictated by the size of the base.

A base greater than $1$. If $r>1$ then $r^x$ is an increasing function.
A base less than $1$. If $0<r<1$ then $r^x$ is a decreasing function.

Here is the graph of $y = 2^x$.

The domain of a core exponential function is all real numbers, $(-\infty , \infty )$.

Graphically, the horizontal axis is a horizontal asymptote.

\[ \lim _{x \to -\infty } 2^x = \answer {0} \, \text { and } \, \lim _{x \to \infty } 2^x = \answer {\infty } \]

When the base is less than $1$, then the behavior switches.

Here is the graph of $y = \left (\frac {1}{3}\right )^x$.

When the base is less than $1$, then the whole function decreases. The function is still unbounded to one side and approaches $0$ on the other.

\[ \lim _{x \to -\infty } \left (\frac {1}{3}\right )^x = \answer {\infty } \, \text { and } \, \lim _{x \to \infty } \left (\frac {1}{3}\right )^x = \answer {0} \]

Note that

\[ \frac {1}{3} = 3^{-1} \]

The function $f(x) = \left (\frac {1}{3}\right )^x$ can be written as

\[ f(x) = \left (\frac {1}{3}\right )^x = \left ( 3^{-1} \right )^x = f(x) = \left ( 3 \right )^{-x} \]

We can view any exponential function with a base less than $1$ as an exponential function with a base greater than $1$ and just change the sign of the exponent.

This is where composition with linear functions comes in.

General Exponential Functions

Our general template for exponential functions looks like

\[ exp(x) = A \cdot r^{B \, x + C} \]

This can be viewd as a composition of a core exponential function and two linear functions.

Let $E(t) = r^t$ be a core exponential function.
Let $L_{in}(u) = B \, u + C$ be a linear function.
Let $L_{out}(v) = A \, v$ be a linear function.

Then, our general exponential function can be expressed as a composition.

\[ A \cdot r^{B \, x + C} = (L_{out}(v) \circ E \circ L_{in})(x) \]

Now we can use the Chain Rule to establish whether an exponential function increases or decreases.

Exponential Function Analysis

Here is the graph of $y = f(x) = \frac {1}{5} \cdot 3^{-2x+1}$.

Domain: The domain of every exponential function is $(-\infty , \infty )$.

Zeros: Exponential functions do not have zeros.

Continuity: Exponential functions are continuous.

End-Behavior: Exponential functions approach $0$ in the direction that makes the exponent negative. Here, the exponent is $-2x+1$, which becomes negative in the positive $x$-direction.

\[ \lim \limits _{x \to \infty } f(x) = 0 \]

In the other direction, exponential functions become unbounded. The sign is given by the sign of the leading coefficient, which is $\frac {1}{5}$, positive.

\[ \lim \limits _{x \to -\infty } f(x) = \infty \]

$f$ can be viewed as a composition of the following component functions.

Behavior: Exponential functions are either increasing or decreasing functions. The end-behavior shows us that $f$ is decreasing.

But, we could also use the Chain Rule.

$f$ is the composition of three functions.

Let $E(t) = 3^t$, an increasing core exponential function.
Let $L_{in}(u) = -2 \, u + 1 $, a decreasing linear function.
Let $L_{out}(v) = \frac {1}{5} \, v$, an increasing linear function.

\[ f(x) = \frac {1}{5} \cdot 3^{-2x+1} = (L_{out} \circ E \circ L_{in})(x) \]

The Chain Rule gives us

\[ inc \circ inc \circ dec = dec \]

Just Checking...This agrees with the graph.

Global Maximum and Minimum: Exponential functions do not have a global minimum or maximum.

Local Maximums and Minimums: Exponential functions do not have local minimums or maximums.

Range: The sign of the leading coefficient tells us that the range is $(0, \infty )$.

Here is the graph of $y = g(m) = -2 \cdot \left ( \frac {2}{3} \right )^{-2x+1}$.

$g$ can be viewed as a composition of the following component functions.

Let $E(t) = \left ( \frac {2}{3} \right )^t$, a decreasing Core exponential function.
Let $L_{in}(u) = -3u+4 $, a decreasing linear function.
Let $L_{out}(v) = -2 \, v $, a decreasing linear function.

\[ g(m) = -2 \cdot \left ( \frac {2}{3} \right )^{-2x+1} = (L_{out} \circ E \circ L_{in})(m) \]

The Chain Rule gives us

\[ dec \circ dec \circ dec = dec \]

$g$ is a decreasing function.

Just Checking...This agrees with the graph.

Global Maximum and Minimum: Exponential functions do not have a global minimum or maximum.

Local Maximums and Minimums: Exponential functions do not have local minimums or maximums.

Range: The sign of the leading coefficient tells us that the range is $(-\infty , 0)$.

Instead of using numbers greater or less than $1$, we could always use a base greater than $1$ and just use negative exponents. In that case, we might as well use $e$ as our base.

This gives us a basic basic exponential function: $e^x$

And, then three other alternative models, if you prefer.

\[ e^x \, \text { or } \, e^{-x} \, \text { or } \, -e^{x} \, \text { or } \, -e^{-x} \]

All exponential functions become unbounded in one direction, while approaching $0$ in the other direction.

When the base is greater than $1$, exponential functions tend to $0$ in the direction that makes their exponent negative.
When the base is less than $1$, exponential functions tend to $0$ in the direction that makes their exponent positive.

Exponential functions are unbounded in the other direction.

Exponential functions can be unbounded positively or negatively. This is determined by the sign of the leading coefficient.

Those are our basic exponential models. Pick one as your own basic exponential function model to memorize.

$exp(x) = e^x$ is very popular.

Then compare all other exponential functions to it.

Graphically

Even though the graphs of exponential functions appear to increase quite quickly, there are no vertical asymptotes. The domains include all real numbers. The function just increases or decreases very quickly and continues to do so.

Here is the graph of $f(x) = -\left ( 3 \right )^{-x + 1}$.

This matches our template for an exponential function.

The leading coefficient is $-1$, which is negative.
The leading coefficient of the exponent is $-1$, which is negative.

Both leading coefficients have the same sign, so this exponential function is increasing.

The leading coefficient of the function is $-1$, which is negative. Therefore, the function only has negative values.

The exponent is $-x+1$. This is negative in the positive tail of the domain. The function tends to $0$ in this direction. The function becomes unbounded in the other direction. Unbounded negatively.

\[ \lim \limits _{x \to \infty } f(x) = 0 \]

\[ \lim \limits _{x \to -\infty } f(x) = -\infty \]

Exponential Analysis

Domain: The domain of every exponential function is $(-\infty , \infty )$.

Zeros: Exponential functions do not have zeros.

Continuity: Exponential functions are continuous.

End-Behavior: In one direction, exponential functions become unbounded. In the other direction, exponential functions tend to $0$.

When the base is greater than $1$,

In the domain, when you move in the direction that makes the exponent unbounded negative, the function tends to $0$.
In the domain, when you move in the direction that makes the exponent unbounded positive, the function is unbounded. Its sign is given by the leading coefficient.

When the base is less than $1$,

In the domain, when you move in the direction that makes the exponent unbounded negative, the function is unbounded. Its sign is given by the leading coefficient.
In the domain, when you move in the direction that makes the exponent unbounded positive, the function tends to $0$.

Behavior (Increasing and Decreasing): Exponential functions are either increasing or they are decreasing.

The Chain Rule can tell you which.

Global Maximum and Minimum: Exponential functions do not have a global maximum or global minimum.

Local Maximums and Minimums: Exponential functions do not have local maximums or minimums.

Range: The range of an exponential function is either $(-\infty , 0)$ or it is $(0, \infty )$.

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2026-05-30 01:33:36