# Exponential function

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)

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 × 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

1. y = 0.5 × 2 x

2. y = -3 × 0.4 x

3. y = ex

4. y = 10x

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