An exponential function is a mathematical function in which the independent variable (x) appears in the exponent. A simple exponential function is represented as $f(x) = a^x$, where 'a' is a constant (the base) and 'x' is the exponent. The value of 'a' must be greater than 0 and not equal to 1. Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. Understanding these functions is crucial for solving problems related to growth and decay processes.
To identify an exponential function, look for a constant base raised to a variable exponent. Linear functions have a constant rate of change (addition or subtraction), while exponential functions have a constant multiplicative rate of change (multiplication or division). For example, $y = 2^x$ is exponential, while $y = 2x$ is linear.
Match the following terms with their definitions:
| Term | Definition |
|---|---|
| 1. Exponential Function | A. The value that is raised to a power |
| 2. Base | B. A function where the independent variable is in the exponent |
| 3. Exponent | C. The number of times the base is multiplied by itself |
| 4. Growth Factor | D. A quantity increasing over time |
| 5. Decay Factor | E. A quantity decreasing over time |
(Match the terms with their definitions)
Complete the following paragraph with the correct terms:
An ________ function has the general form $f(x) = a^x$, where 'a' is the ________ and 'x' is the ________. If 'a' is greater than 1, the function represents exponential ________. If 'a' is between 0 and 1, the function represents exponential ________.
Explain, in your own words, how you can differentiate between a linear function and an exponential function. Provide an example of each.
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