Part 2/5:
Properties of Exponential Functions
Let’s examine some key properties of exponential functions:
- Basic Values:
If ( x = 0 ), then ( a^0 = 1 ).
If ( x = 1 ), then ( a^1 = a ).
For integer ( x ), ( a^n = a^n ), which expands to repeated multiplication (e.g., ( a^2 = a \times a )).
- Rational Numbers:
- If ( x = \frac{p}{q} ) where ( p ) and ( q ) are integers and ( q \neq 0 ), then ( a^{\frac{p}{q}} ) can be expressed as( \sqrt[q]{a^p} ).
- Negative Exponents:
- A negative exponent is defined as ( a^{-x} = \frac{1}{a^x} ). This rule is crucial when dealing with exponential functions.
Graphing Exponential Functions
Graphing exponential functions involves considering several cases based on the value of ( a ):