Exponential function
An exponential function is a function with the general form y = ab^{x} and the following conditions:
 a is a constant and a is not equal to zero (a ≠ 0)
 b is bigger than zero (b > 0)
 b is not equal to 1 (b ≠ 1)
Notice the use of the independent variable (x) as an exponent. This is important in order to have an exponential function.
Why a cannot be equal to 0?
If a = 0, then y = 0 × b
^{x} = 0 since zero times anything is zero.
Therefore, a cannot be zero since it will make the function equal to zero.
Why b must be bigger than zero? In other words, b cannot be zero and b cannot be a negative number.
b cannot be zero since y will be equal to a × 0
^{x} = a × 0 = 0
b cannot be negative either. This can create some problems. For example, suppose b = 1, we get y = a (1)
^{x}
When x = 0.5, y = a(1)
^{0.5} = a √(1) and √(1) is a complex number.
Why b cannot equal to 1?
If b = 1, then y = a × 1
^{x} = a × 1 = a since 1 to any power is equal to 1 and a times 1 is a.
Notice that x disappears as an exponent when b = 1. Therefore, b cannot be equal to 1.
When b > 1, you can model growth and b is the growth factor.
When 0 < b < 1, you can model decay and b is the decay factor.
What does the graph of an exponential function look like?
As you can see from the figure above, the graph of an exponential function can either show a growth or a decay.
The figure on the left shows exponential growth while the figure on the right shows exponential decay.
Examples of exponential functions
1. y = 0.5 × 2
^{x}
2. y = 3 × 0.4
^{x}
3. y = e
^{x}
4. y = 10
^{x}
Can you tell what b equals to for the following graphs?
0.5 × 2
^{x}, e
^{x}, and 10
^{x}
For 0.5 × 2
^{x}, b = 2
For e
^{x}, b = e and e = 2.71828
For 10
^{x}, b = 10
Therefore, if you graph 0.5 × 2
^{x}, e
^{x}, and 10
^{x}, the resulting graphs will show exponential growth since b is bigger than 1.
However, if you graph 3 × 0.4
^{x}, the resulting graph will show exponential decay since b is equal 0.4 and 0.4 is between 0 and 1.

May 26, 22 06:50 AM
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