Exponential Functions, Inverse Functions, Logarithmic Functions,
PSchembari
Exponential Functions
Exponential functions have the independent variable in the exponent. The base is a constant.
General Formula: 𝑦 = 𝑏𝑥 for 𝑏 > 0
Examples: Graph 𝑦 = 2𝑥, 𝑦 = (1
3)𝑥
Depending on the base, these models are usually called exponential growth or exponential decay.
Exponential functions have a horizontal asymptote at 𝑦 = 0 (the 𝑥-axis), and a 𝑦-intercept at (0, 1)
The general graphs, using the general formula, for these functions are
The Important Number 𝒆 and the Important Function 𝒇(𝒙) = 𝒆𝒙
Since the number 𝑒 was first discovered by Napier in 1618, many different formulas have been used to describe it.
Here is one formula to get 𝑒: on your calculator, create a function
𝑦 = (1 +1
𝑥)𝑥
Then look at the values (𝑦-coordinates) when 𝑥 gets really big – over 100,000. You can do this with a table.
For very large 𝑥, this function gets closer and closer to a height of 𝑒 = 2.718281828459… (an infinite decimal which
doesn’t repeat).
On your calculator, you can also get 𝑒 by calculating 𝑒1
A very important exponential function is 𝑓(𝑥) = 𝑒𝑥. Graph it.
In some books and in some software, this is written 𝐸𝑥𝑝[𝑥].
Exponential Growth: b 1
Exponential Decay: 0 b 1
