A primitive function, also known as an antiderivative, is a function whose derivative is the original function. In other words, it is the reverse process of differentiation. The original function is the function that we start with, and the derivative function is the function that we obtain by finding the rate of change of the original function with respect to its variable. In summary, the primitive function is the reverse of the derivative function, and the original function is the starting point for both the primitive and derivative functions. **

The derivative function of the original function is a new function that represents the rate of change of the original function at any given point. It gives us the slope of the tangent line to the original function at each point. In other words, the derivative function tells us how the original function is changing at each point along its graph. **

To assign the derivative function to the original function, you simply take the derivative of the original function. This can be done using differentiation rules such as the power rule, product rule, quotient rule, or chain rule, depending on the complexity of the original function. Once you have found the derivative function, you can then use it to analyze the rate of change of the original function at different points, find the slope of the tangent line, or solve optimization problems. **

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The original function and its inverse function are related in a hierarchical manner. The original function takes an input and produces an output, while the inverse function does the opposite, taking the output of the original function and producing the original input. In this sense, the inverse function "undoes" the original function, making it a sort of "opposite" function. This hierarchical relationship is important in understanding how the two functions are related and how they can be used to solve equations and analyze relationships between inputs and outputs. **

The original function and its inverse function are related in such a way that when one function is applied to the output of the other, the result is the input itself. In other words, if the original function maps an input x to an output y, then the inverse function maps y back to x. This relationship is characterized by the property that the composition of a function with its inverse (or vice versa) results in the identity function. **

The original function is not injective because there are multiple inputs that map to the same output. For example, both 2 and -2 map to 4. However, the original function is surjective because every element in the codomain (positive real numbers) has a preimage in the domain (real numbers). Therefore, the original function is not injective but is surjective. **

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