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Problem 38 Hard Difficulty

Find the domain and range and sketch the graph of the function $ h(x) = \sqrt{4 - x^2} $.

Answer

$[-2,2],[0,2]$

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Video Transcript

for this function f of X. We're going to find the domain range and draw the graph. So remember that the domain is the set of rial number inputs that would yield rial number outputs. And when you're looking at a square root function, you need the number inside the radical to be zero or positive. So first, let's concentrate on what would make it zero. What would make four minus X squared equals zero? That would be true if four equals X squared. And that would be true if X equals plus or minus two. Okay, so to a negative, too could be in the domain. What about numbers greater than two, like three or four. If you had four minus three squared or four minus four squared, then you would get a negative inside your radical and you would get a non riel output. Or if you had negative three for X or negative four for X, the same thing would happen, So we're only going to be able to allow numbers that fall between negative two and two. So the domain is a set of real numbers, from negative to to to and if you want to write it in interval notation. It looks like that with square brackets. Now, how about the range? Well, if you have zero or positive numbers inside the square root, then you're going to get zero or positive numbers for the square root. So the range will be zero to infinity. Let me back up for a second. We started zero, but there actually is a maximum here because the largest that this quantity inside the square root can be the largest. That that can be is for your subtracting zero from four or you're subtracting positive numbers from four. So there's no way that that could get any larger than four. And so the square root of that would be too. So the largest weekend have for a Y value is to Okay, the graph of this turns out to be half a circle. If you've studied the equations circles, you might recognize it. How do we get that graph? There's a number of ways we could plot a bunch of points. We could use a graphing calculator. We could do a combination of those two things

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