What are the asymptotes for `y=3/(x-1)+2` and how do you graph the function?

We are given the rational function `color(green)( f(x) = [3/(x-1)] + 2`

We will simplify and rewrite `f(x)` as

`rArr [3+2(x-1)]/(x-1)`

`rArr [3+2x-2]/(x-1)`

`rArr [2x+1]/(x-1)`

Hence,

`color(red)(f(x) = [2x+1]/(x-1))`

Vertical Asymptote

Set the denominator to Zero.

So, we get

`(x-1) = 0`

`rArr x = 1`

Hence,

Vertical Asymptote is at `color(blue)(x = 1`

Horizontal Asymptote

We must compare the degrees of the numerator and denominator and verify whether they are equal.

To compare, we need to deal with **lead coefficients. **

The lead coefficient of a function is the number in front of the term with the highest exponent.

If our function has a horizontal asymptote at `color(red)(y = a / b)`,

where `color(blue)(a)` is the lead coefficient of the numerator, and

`color(blue)b` is the lead coefficient of the denominator.

`color(green)(rArr y = 2/1)`

`color(green)(rArr y = 2)`

Hence,

Horizontal Asymptote is at `color(blue)(y = 2`

Graph of the rational function with the horizontal asymptote and the vertical asymptote can be found below:

I hope you find this solution with the graph useful.