When you first start learning about functions, you might have to consider them as a machine: You input a value, *x*, into the function, and once it's processed through the machine, another value – let's call it *y* – pops out the far end. The range of possible *x* inputs that can come through the machine to return a valid output is called the domain of the function. So if you're asked to find the domain of a function, you really need to find out which possible inputs would return a valid output.

## The Strategy for Finding Domain

If you're just learning about functions and domains, it's usually assumed that a function's domain is "all real numbers." So when you set about defining the domain, it's often easiest to use your knowledge of mathematics – especially algebra – to determine which numbers *aren't* valid members of the domain. So when you see the instructions "find the domain," it's often easiest to read them in your head as "find and eliminate any numbers that *can't* be in the domain."

In most cases, this boils down to checking for (and eliminating) potential inputs that would cause fractions to become undefined, or have 0 in their denominator, and looking for potential inputs that would give you negative numbers underneath a square root sign.

## An Example of Finding Domain

Consider the function

which really means that any number you input is going to get plopped down in place of *x* on the right-hand side of the equation. For example, if you calculated *f*(4) you'd have

which works out to 3/2.

But what if you calculated *f*(2) or, in other words, input 2 in place of *x*? Then you'd have

which simplifies to 3/0, which is an undefined fraction.

This illustrates one of two common instances that can exclude a number from the domain of a function. If there's a fraction involved, and the input would cause the denominator of that fraction to be zero, then the input must be excluded from the function's domain.

A little examination will show you that absolutely any number *except* 2 will return a valid (if sometimes messy) result for the function in question, so the domain of this function is all numbers except for 2.

## Another Example of Finding Domain

There's one other common instance that will rule out possible members of a function's domain: Having a negative quantity underneath a square root sign, or any radical with an even index. Consider the example function

If *x* ≤ 5, then the quantity underneath the radical sign will be either 0 or positive, and return a valid result. For example, if *x* = 4.5 you'd have

which, while messy, still returns a valid result. And if *x* = −10 you'd have

which, again, returns a valid if messy result.

But imagine that *x* = 5.1. The moment you tiptoe over the dividing line between 5 and any numbers greater than it, you end up with a negative number underneath the radical:

Much later in your math career, you'll learn to make sense of negative square roots using a concept called imaginary numbers or complex numbers. But for now, having a negative number underneath the radical sign rules out that input as a valid member of the function's domain.

So, in this case, because any number *x* ≤ 5 returns a valid result for this function and any number *x* > 5 returns an invalid result, the domain of the function is all numbers *x* ≤ 5.

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About the Author

Lisa studied mathematics at the University of Alaska, Anchorage, and spent several years tutoring high school and university students through scary -- but fun! -- math subjects like algebra and calculus.