Domain of a radical function | Functions and their graphs | Algebra II | Khan Academy
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Domain of a radical function | Functions and their graphs | Algebra II | Khan Academy


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.

8 Comments

  • abhigami

    @mystickybuns1 In this case, greater than or equal to is used because x can equal four, or anything greater than four. If x=4, than we would be taking the square root of zero, which is zero. Less than or equal to cannot be used here because x cannot equal anything less than four, as this would cause a negative number under the square root, giving us an imaginary number as an answer. To answer your question, the equation dictates whether or not to use greater than or equal to or less than/equalto

  • Evey Kaye

    "plain vanilla principle root". I appreciate so much that you explain everything in clear, everyday language that non-math majors can understand! Plus a little humor thrown in! You are so educated, so knowledgeable, but you dont speak like a pompous know-it-all. Your explanations are life-savers for students who are otherwise drowning in "problem-based learning" & inadequate, poorly worded instruction at their schools. THANK YOU!! 

  • Jason Antao

    How come Sal says that the domain is x >= 4. Shouldn't it also be x belongs to real numbers? Also, would the range be y >= 0, because the minimum value is zero? For a function, do we express the domain and range in terms of x and y, or do we just state them. Also, when is it that we have to put the domain and range in the curly brackets?

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