Question

What is the domain and range of `f (x) = (x^2-2)/(x^2-4)`?

Answer 1

Domain and range of this function

Explanation:

Domain:
`x<-2` or `-2 < x < 2` or `x>2`

Range:
`f(x)<= 1/2` or `f(x)>1`

Answer 2

`x inRR,x!=+-2`
`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^2-4=0rArrx^2=4`

`rArrx=+-2larrcolor(red)" excluded values"`

`rArr"domain is " x inRR,x!=+-2`

To find the value that y cannot be find `lim_(xto+-oo)f(x)`

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

`f(x)=(x^2/x^2-2/x^2)/(x^2/x^2-4/x^2)=(1-2/x^2)/(1-4/x^2)`

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

`rArryto1larrcolor(red)" excluded value"`

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

The graph of f(x) illustrates this.
graph{(x^2-2)/(x^2-4) [-10, 10, -5, 5]}