Question

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

Answer 1

`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]}