The domain of an exponential function is all real numbers, and the range is all positive real numbers.
Understanding Domain and Range
- Domain: The set of all possible input values (x-values) for a function.
- Range: The set of all possible output values (y-values) for a function.
Exponential Function Definition
An exponential function is a function of the form:
- f(x) = a^x
Where:
- a is a positive constant called the base.
- x is the exponent.
Domain and Range Explained
Domain:
- Since any real number can be used as an exponent, the domain of an exponential function is (-∞, ∞). This means you can plug in any real number for x.
Range:
- The range of an exponential function is (0, ∞). This means the output of the function will always be a positive number, never zero or negative.
Example
Let's consider the exponential function f(x) = 2^x:
- Domain: (-∞, ∞) - You can input any real number for x.
- Range: (0, ∞) - The output will always be a positive number, never zero or negative.
Practical Insights
Exponential functions are used to model various real-world phenomena, such as population growth, compound interest, and radioactive decay. Understanding their domain and range helps us interpret the results and understand the limitations of the model.
