Free Variable vs. Bound Variable

Instructor: Yuanxin (Amy) Yang Alcocer

Amy has a master's degree in secondary education and has taught math at a public charter high school.

After reading this lesson, you'll be able to identify what free variables and bound variables are. You'll learn the identifying mark for each, so can you know which one has a limitation.

Free Variable

You've already learned about variables and how they can represent almost any number. Well, now, you're going to learn that variables can also be divided into two groups: free and bound. It sounds like one is in jail and the other isn't. This is a good way to describe it, so we will use this comparison throughout the lesson.

A free variable is a variable that has no limitations. It isn't in jail or tied up in any way. It can represent whatever number it needs to represent. Also, the function or expression depends on the free variables. You can say that the value of your free variable determines your answer. When you first started learning about variables, most of them are free variables.

For example, the x in this function is a free variable.

  • f(x) = 3x - 1

Why is it a free variable? It's a free variable because you don't see any limitations put on it. It can equal a 1 or a -1, a 10 or even -1,000. Also, this function depends on this x variable. So, the x here is a free variable.

Bound Variable

A bound variable, on the other hand, is a variable with limitations. It's like it's in jail. Bound variables can't represent whatever number you need it to. Instead, its possible values have already been specified. Also, functions don't depend on its bound variables.

An example of a bound variable is this one.

free and bound variables

Here, the x is the bound variable. This expression specifies that the value of your x goes from 1 to 4 in this summation. Because the x is a bound variable with its values already chosen, the expression isn't dependent on this variable.

Another way you can think about it is that with bound variables, you can swap out the variable for any other variable and the expression will still be the same. For example, if you switched out the bound x in the summation for a w, your summation will still be the same.

free and bound variables

But, if you switched out your free variable for another, then that can totally change your expression. For example, if you switched out the x variable in f(x) = 3x - 1 for an h, that changes your expression.

  • f(h) = 3h - 1

It's different because your x can represent one value while the h represents a totally different value. Your expression depends on these free variables so you can't swap them out.

Identifying Them

A good way to determine whether you are looking at a free or bound variable is to see if your variable has a limitation imposed on it or not. If it does, it's a bound variable. Another criterion is whether or not your expression changes if you swap out the variable for another. If it doesn't, then you have a bound variable. If your function depends on your variable, then you have a free variable.

Take this expression, for example.

  • x2 = 81

Is the variable in this expression free or bound? Let's go through the criteria one by one.

The first criteria say that you have a bound variable if the variable has a limitation. Looking at this expression, you see that the x does indeed have a limitation. According to the expression, the x can only have 9 or -9 value. This means that the x is a bound variable.

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