Determine the domain

spacewater

Junior Member
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Jul 10, 2009
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problem
determine the domains of (a) f\displaystyle f, (b)g\displaystyle g, and (c) fg\displaystyle f \cdot g
f(x)=x+4,g(x)=x2\displaystyle f(x) = \sqrt{x+4} , g(x) = x^2

domain of f - any real number except -4 since the answer will be 0

domain of g - shouldn't the answer be any real number except 0? I don't understand why 0 is included in domain



Can someone elaborate this problem to me please?
 
spacewater said:
problem
determine the domains of (a) f\displaystyle f, (b)g\displaystyle g, and (c) fg\displaystyle f \cdot g
f(x)=x+4,g(x)=x2\displaystyle f(x) = \sqrt{x+4} , g(x) = x^2

f=x+4\displaystyle f=\sqrt{x+4}, domain of f - any real number except -4 since the answer will be 0

The domain are the acceptable values that give a real result. 0 is cool. 0=0\displaystyle \sqrt{0}=0. That's perfectly fine.

What you do not want are negatives inside the radical. Such as 5+4=1=i\displaystyle \sqrt{-5+4}=\sqrt{-1}=i. No good. -5 is NOT in the domain.

Therefore, what is NOT in the domain is (,4)\displaystyle (-\infty, -4)

Everything from negative infinity up to but not including -4 will result in a negative inside the radical. So, the domain is everything but that.

Domain would be [4,)\displaystyle [-4,\infty) because those are the parameters that give us real solutions in the range. The bracket means we include -4. A parenthesis would be

everything up to, but not including, 4. See now?.

Since we can have no negative results from a square root, think of it this way: x+40\displaystyle \sqrt{x+4}\geq 0.

Square both sides. Yes, you can do that. x+40\displaystyle x+4\geq 0

x4\displaystyle x\geq -4. There's the domain. All values of x greater than or equal to -4.

Think of the domain as what goes in and the range as what comes out. The range would be [0,)\displaystyle [0,\infty)

You seem to think that 0 is not allowed. It is. The domain of x2\displaystyle x^{2} would be (,)\displaystyle (-\infty, \infty) because we can plug in anything in the reals for x and get a real result. 02=0\displaystyle 0^{2}=0, (1000)2=1000000\displaystyle (-1000)^{2}=1000000 and so on.
 
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