Exponential function

An exponential function is a function with the general form y  = ab^x 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?

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.

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.