Defining Exponential Functions

For a real-valued exponential function on all real inputs, the base must satisfy:

The restriction $a>0$ keeps $a^x$ real for every real $x$. Excluding $a=1$ separates genuine growth or decay from a constant function.

A scaled exponential function has the form:

In this form, $k\neq0$ is a vertical scale factor. The base controls growth or decay, while $k$ controls the initial value and the side of the $x$-axis on which the graph lies.

Domain Range and Intercept

For the standard function $f(x)=a^x$:

1. The domain is $\mathbb{R}$.
2. The range is $(0,\infty)$.
3. The graph crosses the $y$-axis at $(0,1)$ because $a^0=1$.
4. The graph never crosses the $x$-axis because $a^x>0$.
5. If $a>1$, the function is strictly increasing.
6. If $0<a<1$, the function is strictly decreasing.

For $g(x)=k\,a^x$, the $y$-intercept is $(0,k)$. If $k>0$, its range is $(0,\infty)$. If $k<0$, the graph is reflected across the $x$-axis and its range is $(-\infty,0)$.

Why Two Bases Are Excluded

Base One

If $a=1$, then:

The output never changes, so this is a constant function. Exponential growth and decay require a repeated change by the same factor.

Base Zero

The expression $0^x$ does not define a real-valued function on all real inputs:

At $x=0$, the expression $0^0$ is not assigned a value in this function definition. Therefore, base zero cannot satisfy the required domain $\mathbb{R}$.

Form and Behavior of Exponential Functions

1. $f(x)=4^x$ has $a=4>1$, so it is increasing. Its first values include $f(0)=1$, $f(1)=4$, and $f(2)=16$.

2. $f(x)=3^{x+1}$ can be rewritten as $f(x)=3\cdot3^x$. The factor changes the -intercept from to .

3. $f(x)=5^{2x-1}$ is increasing because the exponent $2x-1$ increases with $x$. For example, $f(0)=5^{-1}=\frac15$ and .

4. $f(x)=0.5^x$ has $0<a<1$, so it is decreasing. Its first values include $f(0)=1$, , and .

Common Models

1. Repeated doubling: $P(t)=P_0\,2^{t/d}$, where $d$ is the doubling time.
2. Compound growth: $A(t)=P(1+r)^t$, where $r$ is the rate per compounding interval.
3. Half-life: $A(t)=A_0\left(\frac12\right)^{t/h}$, where is the half-life.
4. Early transmission: $C(n)=C_0R^n$, where $R$ is an assumed average factor per cycle.

Each formula is meaningful only after its variables, units, interval, and assumptions have been stated. The same exponential shape can describe different contexts, but the symbols do not acquire real-world meaning by themselves.

Published: Apr 2, 2025. Updated: Sep 1, 2026.