How to find the domain of a function?

Published on 27-Sep-2022

The domain is the list of all possible x values that can exist in a function where y= f(x), where x would be the independent variable and y the dependent variable.

If it is a linear equation, quadratic equation, or polynomial equation, the domain will be any real number (-oo+oo)

Example- y=3x+2, y= x2, y= x3+x2+x

If we have a rational function such as y = 2/x-3, as the denominator cannot be 0, the value of x could be anything except for the one that is going to produce 0 in the denominator, x-3≠0 => x≠3, the domain in this would be anything but 3 which can be written as x<3>3(3, oo). Thus, for the rational function, we can find the value of x for which the denominator will be 0 and omit that from the range of the domain.

Some exceptional cases of rational function where y = 2x-3/x2-4; if we consider x2-4≠0 =>x2≠ -4, from here we can say that no matter what, x2 will never be equal to -4 for which the denominator could be 0 hence the domain here would be (-oo, oo) 

In turn of a quadratic function as the denominator such as 2x-3/ x2+3x-12, we can write


=> x2+6x-2x+12≠0

=> (x+6) (x-2) ≠0

=> x≠-6 or x≠2, the domain in this case will be (-oo,-6),(-6,2),(2,oo)

For any radical whose index number is even, such as a square root, there cannot be a negative number on the inside; it must be greater than or equal to 0, Example -/x-4 =>x-4≧0, x≧4. For an odd radical index, the domain could be any real number.

For a quadratic equation in a square root, we find the value of x for which the value on the inside is positive, and that would be our range of domain. Example _/x2+4x-12 => x2+4x-12≧0, x2+6x-2x+12≧0 => (x+6) (x-2) ≧0 => x≦ -6 or x≧2; domain would be (-oo,-6),(2,oo)

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