Why the Multiplication by 11 Trick Works

One of the famous tricks that you have probably known is how to multiply numbers by 11. If we multiply a 2-digit number by 11, we copy the ones digit to the ones digit of the product, add the ones and the tens digit and place the sum in the tens digit of the product, and then copy the tens digit to the hundreds digit of the product. For example, if we multiply 34 × 11.

(1) We copy 4 and place it in the ones digit of the product.
(2) Add 3 and 4 = 7 and place it in the tens digit of the product.
(3) Copy 3 and place it in the hundreds digit of the product.

Therefore, 34 × 11 = 374.  » Read more

Understanding the Concept of Inverse Functions

Let us consider the functions

f(x) = x^3  and g(x) = \sqrt[3]{x}.

The table on the left shows the ordered pairs (x,f(x)). We used f(x) and substitute them to x in the second table.

inverse functions

As we can see, all the values of g(x) are the same as those of x in the first table. For example,  if we have x = 3 in the first table, applying f, we get  27 as output. On the other hand, if we apply g to 27, the value returns to 3» Read more

What exactly is the vertical line test?

A function as we have discussed is a relationship between two sets, where each element in the first set has exactly one corresponding element in the second set. If we think of candies which cost 10 cents each, then we can say that 1 candy costs 10 cents, 2 candies cost 20 cents, 3 candies cost 30 cents, and so on. We can think of this relationship as a function since for each number of candies, there is only one possible price.

If we consider the relation y = 2x, then we can say that it is a function since for every value we substitute to x, there is one and only one corresponding value for y. For instance, if x = -3, then y = -6 and and if x = 9, then y = 18 (one y for each x).  » Read more

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