What Are Functions?

What is y = 3x - 2? An equation? Or maybe a function? Could it possibly be both? Keep on reading to learn all about functions and to find out how they're different from equations.

Jason Marshall, PhD,
February 15, 2014
Episode #186

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IntegralOver the past few weeks, we've talked a lot about functions. In particular, we've been learning all about everyone's favorite trigonometric functions: sine, cosine, and tangent. But, I just realized, we've never actually talked about what makes something a "function" in the first place...until today, that is.

So sit back, relax, and prepare yourself to dive into the wonderful world of functions.

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Quick Review of Equations

For reasons that will soon be apparent, before talking about functions, I'd like to spend a few minutes talking about the equation y = 3x – 2. Why this equation? Well, the most interesting thing about this equation is that unlike most that we've looked at before, this equation has not one but two variables: x and y.

When we first learned about what equations mean, we discovered that we can somewhat fancifully imagine that an equation like 3x - 6 = is something that acts like an old-fashioned scale checking to see if both sides are balanced. For 3x - 6 = x, both sides of the scale equal 3 when x = 3—which means that the scale is balanced for this value of x.

One Variable Equation

Believe it or not, once you understand this idea about the meaning of an equation with one variable, you also understand what an equation with two variables means. By which I mean that just as we can think of a single variable equation as a scale that tests to see if both sides are balanced, we can also think of a two variable equation like y = 3x - 2 as a scale that tests to see if its sides are balanced.

The big difference here is that there isn’t a single number to plug into a variable to see if the equation is balanced. Instead, there are an infinite number of pairs of numbers that will balance the scale. For example, if we set x = 1 in y = 3x - 2, we find that y = 3 • 1 - 2 = 1. If we plug this pair of x and y values into the equation on our scale, we find that it’s balanced since both sides are equal to 1.

Two Variable Equation

While that’s all well and good, there’s actually another way to think about an equation like this with two variables. And, somewhat surprisingly, that way is to not think about it as an equation at all!


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