Question

How do you find the domain and range and intercepTs for `R(x) = (x^2 + x - 12)/(x^2 - 4)`?

Answer 1

see explanation.

Explanation:

To find any `color(blue)"excluded values"` on the domain.

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

`"solve "x^2-4=0rArrx^2=4rArrx=+-2`

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

To find `color(blue)"excluded values"` on the range.

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

`R(x)=(x^2/x^2+x/x^2-(12)/x^2)/(x^2/x^2-4/x^2)=(1+1/x-(12)/x^2)/(1-4/x^2)`

as `xto+-oo,1/x,(12)/x^2,4/x^2to0`

`lim_(xto+-oo),R(x)to(1+0-0)/(1-0)`

`=1/1=1larrcolor(red)" excluded value"`

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

`color(blue)"intercepts"`

`• " let x = 0, in equation, for y-intercept"`

`• " let y = 0, in equation, for x-intercepts"`

`x=0toy=(-12)/(-4)=3larrcolor(red)" y-intercept"`

The numerator is the only part of R ( x ) that can equal zero when y = 0

`rArry=0tox^2+x-12=0`

`rArr(x+4)(x-3)=0`

`rArrx=-4,x=3larrcolor(red)" x-intercepts"`
graph{(x^2+x-12)/(x^2-4) [-10, 10, -5, 5]}