Question

How do you know if an exponential function is increasing or decreasing?

Answer 1

Hint: We know that the general form of the exponential function is $f(x) = {b^x}$ where $b$ must be a positive number, not greater than or equal to zero, and is called the base, and x is called the exponent of the function. So, we use this general form to answer the given question.

Complete step by step solution:
We know that the general form of the exponential function is
 $f(x) = {b^x}$ where $b$ must be a positive number and not greater than one.
Now let us assume that $f(x)$is a function ${D_f}$ as its domain and the range of an exponential function is $\left( {0,\infty } \right)$
Now we take us an example ${f'}(x) = {b^x}$
Here, three cases arise:
1. $x > 0$
2. $x < 0$
3. $x = 0$
Now,
 when $x > 0$ then the overall function is greater than zero,
 when $x < 0$ then we will get a positive quantity in the denominator which is greater than zero,
and when $x = 0$ then we will get a function equal to $1$ which again is greater than zero.
Thus, in all three cases, we came to know that ${f'}(x) = {b^x} > 0$.

Hence, the exponential function is an increasing function

Additional Information: Exponential functions can be used to model things that do not take on negative values and that grow or decay very quickly. We will often see them when looking at things like the number of bacteria in a culture, or in investments that earn compound interest.

Note: The exponential function is a type of mathematical function that is slightly less essential than the linear function. Exponential functions offer the most straightforward solutions to dynamic systems. They can be related to the growth or degradation of a system’s process over time.