About this video
Sal finds the domain of f(x) = sqrt(2x - 8). Because the principal square root is only defined for non-negative numbers, he sets 2x - 8 >= 0 and solves the inequality to get x >= 4.
You'll be able to
- explain what the domain of a function is.
- set up an inequality saying the expression under a square root must be non-negative.
- solve that inequality to state the domain of a radical function.
Key words
- domain
- The set of all input values for which a function is defined.
- principal square root
- The non-negative square root of a number; for real numbers it only exists when the number is 0 or positive.
- non-negative
- Zero or positive, written >= 0.
- inequality
- A statement comparing two expressions with symbols like >, <, >= or <=.
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.
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