# How to Find the Domain of a Function

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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

f(x) = \frac{3}{x - 2}

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

f(4) = \frac{3}{4 - 2}

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

f(2) = \frac{3}{2 - 2}

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

f(x) = \sqrt{5 - x}

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

f(4.5) = \sqrt{5 - 4.5} = \sqrt{0.5}

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

f(-10) = \sqrt{5 - (-10)} = \sqrt{5 + 10} = \sqrt{15}

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:

f(5.1) = \sqrt{5 - 5.1} = \sqrt{-0.1}

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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