Question

What is the domain and range of `f(x)=(x^2-9)/(x^2-25)`?

Answer 1

`x inRR,x!=+-5`
`y inRR,y!=1`

Explanation:

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

`"solve " x^2-25=0rArr(x-5)(x+5)=0`

`rArrx=+-5larrcolor(red)" are excluded values"`

`rArr"domain is " x inRR,x!=+-5`

`"to find any excluded value in the range we can use the"`
`"horizontal asymptote"`

`"horizontal asymptotes occur as"`

`lim_(xto+-oo),f(x)toc" ( a constant)"`

divide terms on numerator/denominator by the highest power of x, that is `x^2`

`f(x)=(x^2/x^2-9/x^2)/(x^2/x^2-25/x^2)=(1-9/x^2)/(1-25/x^2)`

as `xto+-oo,f(x)to(1-0)/(1-0)`

`rArry=1" is the asymptote and thus excluded value"`

`rArr"range is " y inRR,y!=1`