How do you find the domain and range of #1 /( x^3-9x)#?

1 Answer
Jim G.
Jun 6, 2017

#x inRR,x!=0,x!=+-3#
#y inRR,y!=0#

Explanation:

#"the denominator of " y=1/(x^3-9x)" cannot be zero"#

#"as this would make y undefined"#

#"equating the denominator to zero and solving gives the"#
#"values that x cannot be"#

#"solve " x^3-9x=0rArrx(x-3)(x+3)=0#

#rArrx=0,x=+-3larrcolor(red)"excluded values"#

#"domain is " x inRR,x!=0,x!=+-3#

#"to find any excluded values in the range"#
#"consider the horizontal asymptote of the function"#

#"divide terms on numerator/denominator by the highest"#
#"power of x, that is " x^3#

#y=(1/x^3)/(x^3/x^3-(9x)/x^3)=(1/x^3)/(1-9/x^2)#

as #xto+-oo,yto0/(1-0)=0larrcolor(red)" excluded value"#

#"range is " y inRR,y!=0#

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