Definition, domain, and range
Exponential function
An exponential function is any function of the following form:
f\left(x\right)=a^x
a is a positive real number. The number a is called the "base" of the exponential function.
The following function is an example of an exponential function:
f\left(x\right)=2^x
The base of the exponential function is 2.
The domain of an exponential function is all real numbers, \mathbb{R}.
The range of an exponential function is \left(0,\infty\right).
Therefore, if a and x are real numbers with a\gt0, we always have:
a^x\gt 0
The letter e is used to denote Euler's constant. The number e is approximately 2.718.
The exponential function is a function which is important to the study of Calculus and differential equations:
f\left(x\right)=e^x
Graphical representation
Suppose f\left(x\right)=a^x is an exponential function. Then:
f\left(0\right)=a^0=1
Therefore the y -intercept of f\left(x\right) is \left(0{,}1\right).
If a \gt 1, then:
- f\left(x\right) increases to \infty as x tends to \infty.
- As x tends to -\infty the graph of f\left(x\right) approaches the x -axis but never touches it.
If 0\lt a \lt 1, then:
- f\left(x\right) increases to \infty as x tends to -\infty.
- As x tends to \infty the graph of f\left(x\right) approaches the x -axis but never touches it.
If a=1 then a^x=1 for every real number x. Therefore the graph of the exponential function 1^x is the horizontal line y=1.
The following is the graph of the exponential function f\left(x\right)=e^x.
If f\left(x\right)=a^x is an exponential function with a \gt 0 but not equal to 1 then the graph of f\left(x\right) is always above the x -axis, and the range of f\left(x\right) is \left(0,\infty\right).
Properties of exponential functions
Consider the following exponential functions:
- f\left(x\right)=a^x
- g\left(x\right)=b^x
Then, for all real numbers x:
f\left(x\right)g\left(x\right)=a^x b^x=\left(ab\right)^x
Therefore the product of exponential functions is an exponential function.
Consider the following exponential functions:
- f\left(x\right)=2^x
- g(x)= 7^x
Then f\left(x\right)g\left(x\right) is the following exponential function:
f\left(x\right)g\left(x\right)=2^x7^x=\left(2\cdot 7\right)^x=14^x
Consider the following exponential functions:
- f\left(x\right)=a^x
- g\left(x\right)=b^x
Then, for all real numbers x:
\dfrac{f\left(x\right)}{g\left(x\right)}=\dfrac{a^x} {b^x}=\left(\dfrac{a}{b}\right)^x
Therefore the quotient of exponential functions is an exponential function.
Consider the following exponential functions:
- f\left(x\right)=2^x
- g(x)= 7^x
Then \dfrac{f\left(x\right)}{g\left(x\right)} is the following exponential functions:
\dfrac{f\left(x\right)}{g\left(x\right)}=\dfrac{2^x}{7^x}=\left(\dfrac{2}{7}\right)^x
Consider the function:
f\left(x\right)=a^{bx}
a \gt 0 and b is any real number. Then:
a^{bx}=\left(a^b\right)^x
Consider the following exponential function:
f\left(x\right)=3^{2x}
It is equivalent to the following:
f\left(x\right)=3^{2x}=\left(3^2\right)^x=9^x
Consider the following exponential function:
f\left(x\right)=27^{\frac{x}{3}}
It is equivalent to the following:
f\left(x\right)=27^{\frac{x}{3}}=\left(27^{\frac{1}{3}}\right)^x=3^x
Exponential growth and decay
Suppose f\left(x\right)=a^x is an exponential function and a \gt 1. Then as x increases the rate of change of the graph of f\left(x\right) is increasing.
The rate at which f\left(x\right) increases is quite drastic and is referred to as exponential growth.
Suppose f\left(x\right)=3^x. The following table lists the values of f\left(x\right) for the first ten natural numbers:
| x | 3^x |
|---|---|
| 1 | 3 |
| 2 | 9 |
| 3 | 27 |
| 4 | 81 |
| 5 | 243 |
| 6 | 729 |
| 7 | 2187 |
| 8 | 6561 |
| 9 | 19 683 |
| 10 | 59 049 |
The following graph is a zoomed out image of the graph of f\left(x\right)=3^x. Notice that we can only plot points for the x -values of 0, 1, 2 and 3. If the image were zoomed out further we could also plot the point \left(4{,}81\right), then the other points would be indistinguishable from one another.
Similarly, as x decreases towards -\infty the graph of f\left(x\right) becomes arbitrarily close to the x -axis. The rate that the graph approaches the x -axis is referred to as an exponential decay.