SOLUTION: what is the domain and range of f(x)=x^2/(4-x^2)
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Question 484124: what is the domain and range of f(x)=x^2/(4-x^2)
Answer by lwsshak3(11628) (Show Source): You can put this solution on YOUR website!
what is the domain and range of f(x)=x^2/(4-x^2)
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Domain is a set of all possible inputs (x-values) for which the function is defined.
Range is a set of all possible outputs (y-values)
f(x)=x^2/(4-x^2)
y=x^2/(4-x^2)
Domain:
F(x) is undefined when the denominator=0.
To find the domain, set the denominator =0, then solve for x.
4-x^2=0
(2-x)(2+x)=0
2-x=0
x≠2
and
2+x=0
x≠-2
Domain: (-∞,-2) U (-2, 2) U (2, ∞)
..
Range:
The range in many instances can easily be determined by inspection, but in this case, it is not that obvious. The procedure to use is to solve for x in terms of y. What you are doing, in effect, finding the inverse of the function where domain and range are interchanged, that is, the domain of the function is the range of the inverse and vise=versa.
..
y=x^2/(4-x^2)
4y-x^2y=x^2
4y=x^2y+x^2=x^2(y+1)
x^2=4y/y+1
x=±√(4y/y+1)=±√4y/√y+1
4y≥0
y=≥0
..
y+1≥0
but y+1≠0 (because it is a denominator)
y+1>0
y>-1 (This is also not within range because in the interval (-1, 0), the radican, (4y/y+1) would become negative)
Range, therefore: (-∞, -1) U [0, ∞)
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