Student: So now I know all about building functions with variables! This isn't so bad!
Mentor: Well, what you have actually learned so far is what a simple one operation functions is like.
This is just a small part of a bigger picture. The next type of function we should look at is
a multi-step function. Do you think you can give me an example of a multi-step function?
Student: Let me think. What about taking a number, adding 1 to it and then multiplying by 3?
Mentor: That's right! Now how would you write that using the variables x and y?
x + 1 * 3 = y ?
Mentor: Sort of. We need to be a little careful here, so that the person using the function knows to
add first and then multiply. Mathematicians realized a long time ago (in the fifteenth century
as algebra was being used more and more) that there needed to be rules about how to write
anything with more than one operation so there would be no ambiguity. A standard evolved for
the order of operations, which we still follow today:
Do all parentheses first
All exponents next
Then comes multiplication and division from left to right
Save all additions and subtractions for last and compute those left to right
Student: So for clarity I need to write it as
y = (x + 1) * 3
Mentor: Yes. You can also write the same equation these other ways:
y = 3 * (x + 1)
y = 3(x + 1)
y = 3 * x + 3
y = 3x + 3
Student: Well, each one gives the same output using 3 as an input.
Mentor: That's true, but we have seen functions that are different for some numbers and the same for
others. For example, y = x + 6 and y = 3 * x both give 9 as an output when 3 is the input. Can
you explain it another way?
Student: Well, for the functions you just named, the output is only the same for the input 3, where
the functions you gave earlier are the same for
Mentor: Good observation. So they are all the same because they are rearrangements that follow
arithmetic properties. These properties say:
We can multiply or add two numbers in any order. This is the commutative property.
We can distribute multiplication across addition. This is the distributive property.
Student: So the same function can have different forms?
Mentor: Exactly - and when we allow more complicated things like lots more than just one or two
operations and include exponentiation in the list, checking that two functions are equivalent
gets more challenging.