Question

How do you graph `g(x) = -9/(x-9)+6`? What is the domain and range?

Answer 1

graph{(6x-63)/(x-9) [-75.7, 84.3, -37.2, 42.8]}

domain : `R-{9}`
range : `R-{6}`

Explanation:

simplify the function : `g(x)=6-(9/(x-9))=((6(x-9))/(x-9))-9/(x-9)`

`=(6x-54-9)/(x-9)=(6x-63)/(x-9)`

the function has horizontal asymptote at `y=6`

and vertical asymptote at `x=9`

so the `x` never reaches 9, `y` never reaches 6

Answer 2

see explanation.

Explanation:

The denominator of g(x) cannot be zero as this would make g(x) `color(blue)"undefined".` Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

`"solve " x-9=0rArrx=9" is the asymptote"`

`rArr" domain is " x inRR,x!=9`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

`g(x)=-(9/x)/(x/x-9/x)+6=-(9/x)/(1-9/x)+6`

as `xto+-oo,g(x)to-0/(1-0)+6`

`rArry=6" is the asymptote"`

`rArr"range is " y inRR,y!=6`

`color(blue)"Intercepts"`

`x=0toy=-9/(-9)+6=7larrcolor(red)" y-intercept"`

`y=0to-9/(x-9)+6=0`

`rArr9/(x-9)=6rArrx=21/2larrcolor(red)" x-intercept"`
graph{-(9/(x-9))+6 [-25.66, 25.65, -12.83, 12.83]}