What is the domain and range of the function #f(x) = x/(x^2-4)# ?

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Answer 1 · 2018-01-18T13:17:37

The domain is #x in (-oo,-2)uu(-2,2)uu(2,+oo)#.
The range is #y in RR#

The function is #f(x)=x/(x^2-4)=x/((x+2)(x-2))#

As the denominator must be #!=0#, so

#x!=2# and #x!=-2#

The domain is #x in (-oo,-2)uu(-2,2)uu(2,+oo)#

To find the range, proceed as follows :

#y=x/(x^2-4)#

#y(x^2-4)=x#

#yx^2-4y=x#

#yx^2-x-4y=0#

In order for this quadratic equation in #x# to have solutions, the discriminant #Delta>=0#

#b^2-4ac=(-1)^2-4(y)(-4y)>=0#

#1+16y^2>=0#

Therefore,

#AA y in RR,#, #Delta>=0#

The range is #y in RR#

graph{x/(x^2-4) [-10, 10, -5, 5]}