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.

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 × 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.

What does the graph of an exponential function look like?

Exponential function

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 × 2x

2. y = -3 × 0.4x

3. y = ex

4. y = 10x

Can you tell what b equals to for the following graphs?

0.5 × 2x, ex, and 10x

For 0.5 × 2x, b = 2
For ex, b = e and e = 2.71828
For 10x, b = 10

Therefore, if you graph 0.5 × 2x, ex, and 10x, the resulting graphs will show exponential growth since b is bigger than 1.

However, if you graph -3 × 0.4x, the resulting graph will show exponential decay since b is equal 0.4 and 0.4 is between 0 and 1.

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