Question

If the base of the exponent were 1, the function would not change over time because any number raised to the power of 1 equals the original number. If the base of the exponent were between 0 and 1, the function would still represent exponential decay, but at a slower rate compared to when the base is greater than 1.

Answer 1

That's correct! Let's break this down further:

1. Base of 1: When the base of an exponential function is 1 (i.e., \( f(x) = 1^x \)), the output is always 1 regardless of the input \( x \). This means there is no change over time; it's a constant function.

2. Base between 0 and 1: If the base of the exponential function is between 0 and 1 (e.g., \( f(x) = a^x \) where \( 0 < a < 1 \)), the function represents exponential decay. As \( x \) increases, the value of the function \( f(x) \) decreases, but at a slower rate. The output approaches zero but will never actually reach it. This behavior is characteristic of decay processes, such as radioactive decay or depreciation of value over time.

3. Base greater than 1: Conversely, when the base is greater than 1 (e.g., \( f(x) = b^x \) where \( b > 1 \)), the function represents exponential growth. As \( x \) increases, the value of the function increases rapidly.

In summary, the base of the exponent crucially determines whether the function shows constant behavior, exponential decay, or exponential growth, and affects the rate at which these changes occur.