Final answer:
The function f(x) where f(x) is equal to f'(x) is f(x) = e^x. The keyword here is 'exponential,' as only this type of function maintains the property that the function value equals its rate of change with respect to x.
Step-by-step explanation:
The question you're asking essentially can be broken down into finding a function f(x) such that it is equal to its own derivative, or in other words, f(x) = f'(x). This question is asking you to identify functions where the output is equal to the rate at which the function's value changes with respect to x.
From the options and notes you provided, the function that represents an exponential growth pattern like the one described is f(x) = e^x, where e is the base of the natural logarithm. This is because the derivative of e^x with respect to x is also e^x, which means that the function is equal to its derivative at all points.
Regarding the other provided information, it seems there might be some confusion or typos, so it's important to ensure the question is clear before attempting to answer further.