Eli J. answered • 02/16/20
Expert Math Tutor: All Ages and Courses
I'm going to do this two ways: first, the uncreative straightforward way that is a bit of work, but not difficult. Second, the creative way where with a little bit of thinking you can get the answer in a few seconds. The reason I'm going to describe it this way is that very often when you get help with a problem, the helper will go straight to the "clever" solution. But if you didn't come up with the clever way it can sometimes be a little discouraging; you might think, "I didn't see the clever way, I'm not good at this kind of stuff."
The secret is this: you always start out by doing new types of problems the brainless, "automatic" way where you're just following the rules. But if you spend some time thinking about the problem after you solve it, you can sometimes see the patterns you missed the first time. Do this enough, and the "clever" way will just come to you without too much work.
Ok, here goes: f(3) means take f(x) and plug in 3 wherever we see x. So
f(3) = 12*39 + 3 = 236196 + 3
f(3) = 236199
Likewise, f(-3) means take f(x) and plug in (-3) wherever we see x. So
f(-3) = 12*(-3)9 + (-3) = -236196 - 3
f(-3) = -236199
Putting those two things together, we get
f(3) + f(-3) = 236199 - 236199 = 0
We're done already, but we won't learn anything if we stop here. Let's think about this a bit. It turns out that f(3) was the exact opposite of f(-3). Was this a coincidence? If you try a few other values for x, you will see this wasn't a coincidence. For instance, f(1) is the opposite of f(-1). If we generalize this rule, we can say that for any value of x,
f(x) = -f(-x)
A function that has this behavior is called an odd function. If your function is a polynomial (i.e. the terms are whole number powers of x, with coefficients), then there is a way to tell just from looking at the powers whether the function is odd or not! All you have to do is look at the powers... if they are all odd, then the function is odd! In this case, one term has an exponent of 9, and the other term has an exponent of 1. Both 9 and 1 are odd, so this function is odd. Which means you can look at it right away and know that f(3) is the opposite of f(-3). If you add opposites you always get zero, so if you recognized this function was odd, you would be able to look at it and instantly know the answer is zero, without doing any complicated exponents on your calculator.
