What is the domain and range of #g(x) =(5x)/(x^2-36)#?

1 Answer
Feb 10, 2018

Answer:

#x inRR,x!=+-6#
#y inRR,y!=0#

Explanation:

The denominator of g(x) cannot be zero as this would make g(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be.

#"solve "x^2-36=0rArr(x-6)(x+6)=0#

#rArrx=+-6larrcolor(red)" are excluded values"#

#rArr"domain is "x inRR,x!=+-6#

#"or in interval notation as"#

#(-oo,-6)uu(-6,6)uu(6,+oo)#

#"for range divide terms on numerator/denominator by the"#
#"highest power of x that is "x^2#

#g(x)=((5x)/x^2)/(x^2/x^2-36/x^2)=(5/x)/(1-36/x^2)#

#"as "xto+-oo,g(x)to0/(1-0)#

#rArry=0larrcolor(red)"is an excluded value"#

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

#(-oo,0)uu(0,+oo)larrcolor(blue)"in interval notation"#
graph{(5x)/(x^2-36) [-10, 10, -5, 5]}