1.4.2 Additive Functions

Here's an important algebraic question about functions for which the answer has to be a whole class of functions: What functions are additive?

Example 1

Consider the function whose rule is “multiply by 5.” Is this function additive?

Solution   In symbols, this function is defined by the formula

We calculate that

so the multiply-by-5 function is indeed additive.

Here is a procedure for deciding whether a function is additive or not:

1. If you suspect the function is not additive, choose particular numbers `a` and `b` in the domain of the function, and calculate $f(a+b)$ and $f\left(a\right)+f\left(b\right)$. If these two numbers turn out to be different, you have shown that the function is not additive. If they turn out to be the same, try two other numbers `a` and `b`.

2. On the other hand, if you suspect the function is additive, try to show that by an algebraic calculation like the one in Example 1.

Activity 2

1. For each of the following functions, decide whether the function is additive or not. Write a one-sentence reason for your conclusion.

   (i) ${\displaystyle f\left(x\right)=-2x}$		(ii) ${\displaystyle f\left(x\right)=-2x+7}$		(iii) ${\displaystyle f\left(x\right)=x^2}$
   (iv) ${\displaystyle f\left(x\right)=\frac{2}{x}}$		(v) ${\displaystyle f\left(x\right)=\sqrt{x}}$		(vi) ${\displaystyle f\left(x\right)=\log \left(x\right)}$

2. Describe the largest class of functions you can think of that you know for sure are all additive. How do you know for sure?

3. Would you describe additive functions as “relatively common” or “relatively rare”?

Comment on Activity 2

The additivity condition, $f(a+b)=f\left(a\right)+f\left(b\right)$, is an example of a functional equation, that is, an equation involving an unknown function that might or might not be satisfied by any particular function. Your work on Activity 2 was an attempt to separate functions that satisfy this equation from functions that don't.

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