1.4.3 Multiplicative Functions

Example 2

Is the multiply-by-5 function multiplicative?

Solution   Let's try two numbers in its domain, say $a=2$ and $b=3$. Then $f\left(a\right)=5\cdot 2=10$, and $f\left(b\right)=5\cdot 3=15$, so $f\left(a\right)f\left(b\right)=150$. On the other hand, $f(ab)=f\left(6\right)=30$, so this function is definitely not multiplicative.

Activity 3

1. For each of the following functions, decide whether the function is multiplicative 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 multiplicative. How do you know for sure?

3. Would you describe multiplicative functions as "relatively common" or "relatively rare"?

Comment on Activity 3

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