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Suppose $f$ is not a constant function with $f(x+y)=f(x) f(y),$ show (a) $f(0)=1$,(b) $\frac{f(x+h)-f(x)}{h}=f(x)\left(\frac{f(h)-1}{h}\right)$.

Algebra

Chapter 1

Functions and their Applications

Section 7

More on Functions

Functions

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McMaster University

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Lectures

01:43

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x^2. The output of a function f corresponding to an input x is denoted by f(x).

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for part of a we need to use the condition given in the question f x plus y equals f x times f y. First we assume that both X and y are zeros, so f zero equals F zero plus zero. According to the about condition, these echoes f zero times F zero, which is f zero squired. So if this equation held after Darryl counting be there or what next we assume y equals zero. So f X equals f X plus zero, which is f x times after if F zero equals zero. Yes, we all gave f X equals zero. But the question also says that F is not a counseling function. This need to a contradiction, so f X cannot be zero. And, uh, F zero is now zero. So if there are can only be what which gives the result we want for per p. We still use a condition f x plus h equals F s times f h. Then we replace this condition into the expression we want and with some very simple calculations, we will get the result

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