How To Find The Domain Of A Function

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.

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.

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.

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.