A function is a process that accepts inputs, performs some operations on them, and then gives out outputs. It’s quite similar to production lines where raw materials are fed to a machine, and they come out on the other end as finished products. 

Cotton candy is actually a perfect example. Sugar is fed into a revolving machine, and the centrifugal forces convert it into long fluffy filaments. But what happens when water is fed into the machine? Or a handful of sand? 

Clearly, some inputs mess up the operation of the machine to the point that it doesn’t work. It’s imperative then to set apart the inputs that produce valid results from the ones that don’t. In other words, it’s important to learn how to find the domain of a function. 

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Any function y=f(x) is defined by the various values of y for each input x. 

In general, functions should work over the whole range of real numbers, and this happens quite often for simple linear and quadratic functions. 

Are both functions where all inputs produce valid outputs. 

However, there are some exceptions where functions include limitations and discontinuities. 

Maintaining the Positivity of Square Roots 

Here, the value of z can be any number that equals or exceeds -9, otherwise, the square root would be a negative entity. This is unacceptable in the realm of regular algebra. It would be permissible though in an imaginary or complex numbers domain.

The restrictions here come from the fact that the denominator cannot be zero at any time. Dividing by zero is undefined, and that’s why the function k(x) wouldn’t work whenever x takes on the value of 3.

The domain is therefore the set of all inputs that produce valid outputs in a function. 

An alternative way to look at the domain would be defining the range of values where the function works within the limitations of real numbers.

What Exactly Is the Domain?

The beauty of mathematical tools is that they’re accurate, easy to follow, and they give consistently correct outcomes. That’s as long as a simple procedure is followed.   

The problem at hand could be modeling the performance of an assembly line at a car factory, assessing the efficacy of a certain medicine, or just a hypothetical algebraic exercise. For all these functions there’s a workable domain. And here’s how to find it.   

This is a quick and easy way to assess the domain of a function. A rough estimate could be sufficient, but often an exact analysis of the figures is necessary. 

The following function has a peculiar shape that’s worth checking:

From a quick look at the graph, it’s obvious that the domain is all the values from -2, including that point, all the way to just before 3. And then, right after 3, all the way to infinity. 

To express this result in a more professional and succinct manner: 

Step 2: Find the Domain From the Coordinates of a Function

A function could be defined by a set of coordinates where an input xhas a unique output y. 

In that case, the domain is simply the set of all the x values. 

This simply means that the function is only defined at these specific points. Outside of these values, there’s no clear expectation of how it would perform. Thus, it’s only safe to use that specific function with inputs 1,3,5, and 7. 

The function could function smoothly from negative infinity to positive infinity, except for a couple of values wherein it goes totally out of whack. This is called a discontinuity. 

To find the domain of a function, it’s then wise to look for all the spots where the outcome wouldn’t be as expected. 

The number line often helps in identifying such spots or ranges. 

This function would work great for every value that’s not 2. The number line would show a couple of rays drawn left and right around the number 2, but excluding it. In the representation, the number two would have a blank circle on it.

The domain could also be expressed as (- ,2)  (2, ) 

Some mathematical functions aren’t defined under certain conditions. Dividing by zero, finding the log or a square root of a negative number, or calculating tan (90°)

This takes some knowledge of the performance of various functions. Otherwise, who could guess that there’s an impossible value in its domain? 

Step 5: Express the Results and State the Domain 

The domain of a function is typically expressed as an interval, union of intervals, or a set. In addition, the boundaries of intervals can be represented in two different ways. The first uses an inverted square bracket to represent open intervals, and the second uses a curved bracket.

To demonstrate; a domain of all the numbers smaller than 2, but excluding 2, and all the numbers bigger than 4, including 4, would look like this: 

Which one to use for expressing the limitations of a function? This depends to a large extent on the notation used in each country. If there are no specific requirements regarding that notation, then it’s generally accepted to choose what feels more convenient.

How to Find the Domain of a Function

Math is best understood if the various concepts are mapped to real-life applications. That’s also how these calculations help in creating innovative devices, algorithms, codes, and even in making sound business decisions.    

That was the easy method of how to find the domain of a function. The next step is often finding the range, which is the corresponding output for the permissible input values.

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How To Find the Domain of a Function

What Exactly Is the Domain?

How to Find the Domain of a Function

In Conclusion