Question

The ordered pairs (1, 1), (2, 4), (3, 9), (4, 16), and (5, 25) represent a function. What is a rule that represents this function?

Answer 1

You can think of a function sort of like a box where you put something in and it comes out the other side different but with a consistent pattern. So, in this case, 1 goes in and comes out 1; 2 goes in and comes out 4; 3 goes in and comes out 9; 4 goes in and comes out 16; 5 goes in and comes out 25. We know that there is definitely a pattern, because we are told it is a function, but what is the pattern?
So, first we can look at addition. 1 goes in, we add 0 to it to get 1. 2 goes in, if we add 0 to 2, we get 2, which is incorrect.
So, now we look at multiplication/division. 1 goes in, we multiply by 1 and get 1. 2 goes in, we multiply it by 1, and get 2... so nope, no answer there either.
So, now what else is there to look for? Usually next in these easier patterns you will look at exponential. So, 1 goes in, 1 to the power of anything is 1. Now lets look at the 2nd one. 2 to what power is 4? 2 to the power of 2 is 4. Let's check the next one. 3 to the power of 2 is 9... so the pattern seems to work. If we test every entry, we will find that the pattern is that whatever goes in is square to get the result.
Make sense? The more algebra you do, the better grasp you will get and the faster you will become. Good luck! =)