Question

How do you graph `f(x)=(-x-1)/(x+2)` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

see explanation.

Explanation:

`color(blue)"Asymptotes"`

The denominator of f(x) cannot be zero as this would make f(x) 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+2=0rArrx=-2" is the asymptote"`

Horizontal asymptotes occur as

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

divide terms on numerator/denominator by x

`f(x)=(-x/x-1/x)/(x/x+2/x)=(-1-1/x)/(1+2/x)`

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

`rArry=-1" is the asymptote"`

Holes occur if there is a duplicate factor on the numerator/denominator. This is not the case here hence there are no holes.

`color(blue)"Intercepts"`

`x=0toy=-1/2rArr(0,-1/2)larr"y-intercept"`

`y=0to-x-1=0tox=-1rArr(-1,0)`
graph{(-x-1)/(x+2) [-10, 10, -5, 5]}