Question

How do you graph `f(x)=4/(x-1)+1` using holes, vertical and horizontal asymptotes, x and y intercepts?

Answer 1

see explanation.

Explanation:

We can express f(x) as a single rational function.

`rArrf(x)=4/(x-1)+(x-1)/(x-1)=(x+3)/(x-1)`

`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-1=0rArrx=1" 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+3/x)/(x/x-1/x)=(1+3/x)/(1-1/x)`

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

`rArry=1" is the asymptote"`

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

`color(blue)"Approaches to vertical asymptote"`

`lim_(xto1^-)=- oo" and " lim_(xto1^+)=+oo`

`color(blue)"Intercepts"`

`x=0tof(0)=3/(-1)=-3larr" y-intercept"`

`y=0tox+3=0tox=-3larr" x-intercept"`
graph{(x+3)/(x-1) [-10, 10, -5, 5]}