Question

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

Answer 1

`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`