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## Algebra (all content)

### Course: Algebra (all content) > Unit 12

Lesson 4: Domain of radical functions# 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?

- 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)- a radicle funtion is one which includes radical sign i.e square root(17 votes)

- 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)
- 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)

- 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)
- 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)

- What is the principle square root? Is it different from the square root?(3 votes)
- 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)

- what is the domain of √|x|-x(1 vote)
- √|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)

- Around0:47he mentions a "plain vanilla one for real numbers." Does that mean there is a square root for negative numbers?(3 votes)
- There is a square root for negative numbers but it is not yet taught at this level of math.(3 votes)

- 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)
- 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)

- how come x can be equal to 4? isn't the square root of zero is undefined?(1 vote)
- √0 = 0 because 0 • 0 is defined, but 0/0 is not.(4 votes)

- 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)
- Yes, your understanding is correct.

The range would be from any real number greater than zero. This range encompasses all non-negative real numbers.

Cheers.(2 votes)

- 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)
- 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.