If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Domain of a radical function

Finding the domain of f(x)=√(2x-8). Created by Sal Khan and Monterey Institute for Technology and Education.

Want to join the conversation?

  • mr pants teal style avatar for user Jahnavi Sunkara
    Hey guys, Janu here!
    So I have one major question that might or might not be simple to answer. So here it goes:

    What is a radical function?

    Thank in advance for answering,
    Respond ASAP
    (8 votes)
    Default Khan Academy avatar avatar for user
  • mr pants teal style avatar for user Hollerdog
    Does anyone else get f(x) belongs to all real numbers where f(x) is greater than or equal to 0, for the range?
    (6 votes)
    Default Khan Academy avatar avatar for user
    • mr pink red style avatar for user Matthew (Cole) Brogden
      Because whenever you use 1 or something for x you get something less then 8 and since you are subtracting 8 you would get a negative number, which is a no, no. so X has to be greater then or equal to 4 because 2(4) = 8 and 8-8 = 0 so 4 is the lowest possible answer, thus the domain is anything greater then or equal to 4
      (7 votes)
  • blobby green style avatar for user Amol Acharya
    In the exercise, what does it mean to say that a piecewise defintion applies when x is not equal to 8? Thank you.
    (6 votes)
    Default Khan Academy avatar avatar for user
    • orange juice squid orange style avatar for user Charles Pensacola
      A piecewise function works like other functions when that given an x value the function gives back an answer y. The difference in a piecewise function is that you have to determine which equation to use to calculate the y value. For example, let's assume we have a piecewise function with two equations; equation A: y=12 and equation B: y=2x+2). If x is equal to 8 then use equation A to calculate y and is this case for x=8, y will equal 12. If x is not equal to 8 (such as 7 or 10.2 or any other value that's not 8) then use equation B to calculate y. For example, if x=6, then we would plug 6 into equation B for x. y=2(6)+2 which would result in y=14
      (5 votes)
  • duskpin ultimate style avatar for user Ashton Tang
    What is the principle square root? Is it different from the square root?
    (3 votes)
    Default Khan Academy avatar avatar for user
    • leaf green style avatar for user kubleeka
      The square root of a number x is any number that, when you multiply it by itself, gives you x. So a square root of 4 is 2, since 2•2=4.

      Now note that (-2)•(-2) also equals 4. So 4 has two square roots. (In fact, so does every number except 0.)

      But sometimes, we want square rooting to give only one answer, like in geometry or if we want √x to be a function. So we can also take the principal root, also denoted √x, which is just the positive square root of x.
      (10 votes)
  • male robot johnny style avatar for user Anshul Kumar
    what is the domain of √|x|-x
    (1 vote)
    Default Khan Academy avatar avatar for user
    • cacteye blue style avatar for user Jerry Nilsson
      √|𝑥| is defined for all 𝑥, such that |𝑥| ≥ 0, which is true for all 𝑥 ∈ ℝ.
      −𝑥 is of course defined for all 𝑥 ∈ ℝ.

      So, √|𝑥| − 𝑥 is defined for all 𝑥 ∈ ℝ, and thereby the domain is 𝑥 ∈ (−∞, ∞)

      – – –

      √(|𝑥| − 𝑥) is defined for all 𝑥, such that |𝑥| − 𝑥 ≥ 0 ⇔ |𝑥| ≥ 𝑥, which is true for all 𝑥 ∈ ℝ.
      Again, the domain is 𝑥 ∈ (−∞, ∞)
      (9 votes)
  • leaf blue style avatar for user eabmath
    Around he mentions a "plain vanilla one for real numbers." Does that mean there is a square root for negative numbers?
    (3 votes)
    Default Khan Academy avatar avatar for user
  • hopper jumping style avatar for user Mark Ivanovich
    But, what about complex number?I am here 'couse this video included in unit Algebra 2 (Domain of radical functions) and complex numbers are already learned.
    (3 votes)
    Default Khan Academy avatar avatar for user
    • male robot hal style avatar for user Alexander Bessonov
      Good question, Mark. The domain is usually defined for the set of real numbers that can serve as the function's input to output another real number. If you input any number less than 4, the output would be a complex number, and would not count toward the domain. The function provided in the video would be undefined for real numbers less than 4.

      Additionally, if you wanted to find a complex domain for the function in the video, it would at least partially consist of all complex numbers, since for that function, any complex number you put in it will also output another complex number.

      If you are interested in this topic, there is a branch of mathematic dealing with functions with complex numbers called complex analysis. The domains of the functions they use usually cannot be expressed be expressed linearly, as the functions have two outputs (real output and complex output). Many of these functions express a domain which occupy two dimensional spaces, including the complex plane.
      (1 vote)
  • winston baby style avatar for user SangHyun Hwang
    how come x can be equal to 4? isn't the square root of zero is undefined?
    (1 vote)
    Default Khan Academy avatar avatar for user
  • leafers ultimate style avatar for user Elaine Nagahara
    Would the range be all real non-negative numbers? When you plug in a 4 for x, you end up getting the sqrt. of 0 which is 0. So I assumed any number higher than 4 (x>=4) would result in a larger sqrt. than 0. Is this correct?
    (2 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Ayshi
    Why when n is odd the domain of a radical function is defined for all real numbers and when n is even the domain is solely defined for all non-negative real numbers?
    (2 votes)
    Default Khan Academy avatar avatar for user
    • primosaur seed style avatar for user Ian Pulizzotto
      In the real number system, it is possible to find odd roots, but not even roots, of negative numbers. This is a consequence of the fact that a negative number to an odd power is negative, but a negative number to an even power is positive.

      Have a blessed, wonderful day!
      (2 votes)

Video transcript

Find the domain of f of x is equal to the principal square root of 2x minus 8. So the domain of a function is just the set of all of the possible valid inputs into the function, or all of the possible values for which the function is defined. And when we look at how the function is defined, right over here, as the square root, the principal square root of 2x minus 8, it's only going to be defined when it's taking the principal square root of a non-negative number. And so 2x minus 8, it's only going to be defined when 2x minus 8 is greater than or equal to 0. It can be 0, because then you just take the square root of 0 is 0. It can be positive. But if this was negative, then all of a sudden, this principle square root function, which we're assuming is just the plain vanilla one for real numbers, it would not be defined. So this function definition is only defined when 2x minus 8 is greater than or equal to 0. And then we could say if 2x minus 8 has to be greater than or equal to 0, we can solve this inequality to see what it's saying about what x has to be. So if we add 8 to both sides of this inequality, you get-- so let me just add 8 to both sides. These 8's cancel out. You get 2x is greater than or equal to 8. 0 plus 8 is 8. And then you divide both sides by 2. Since 2 is a positive number, you don't have to swap the inequality. So you divide both sides by 2. And you get x needs to be greater than or equal to 4. So the domain here is the set of all real numbers that are greater than or equal to 4. x has to be greater than or equal to 4. Or another way of saying it is this function is defined when x is greater than or equal to 4. And we're done.