An exponential function is a function with the general form y = abx and the following conditions:
x is a real number
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)
Why a cannot be equal to 0?
Notice the use of the independent variable (x) as an exponent. This is important in order to have an exponential function.
If a = 0, then y = 0 × bx
= 0 since zero times anything is zero
Therefore, a cannot be zero since it will make the function equal to zero.
b must be bigger than zero.
In other words, b cannot be zero and b cannot be a negative number.
b cannot zero since y = a × 0x
= 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 × 1x
= 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.
Examples of exponential functions
y = 0.5 × 2 x
y = -3 × 0.4 x
y = ex
y = 10x
Oct 22, 18 04:14 PM
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