# How do you find the domain and range of # f(x) = (x+7) / (x-5)#?

##### 1 Answer

I got a domain and range of:

#(-oo, 5) uu (5, oo)# , or#x ne 5#

#(-oo, 1) uu (1, oo)# , or#y ne 1#

The function is undefined for *denominator*,

#f(5) = (5 + 7)/(5 - 5) = color(green)(12/0)#

Since the domain is based on the allowed values of **domain** is:

#color(blue)((-oo,5) uu (5,oo))#

Based on the domain, we would find the range by solving for

#y = (x + 7)/(x - 5)#

#y(x-5) = x + 7#

#xy - 5y = x + 7#

#x - xy = -5y - 7#

#x(1 - y) = -5y - 7#

#x = (-5y - 7)/(1 - y)#

#color(green)(x = (5y + 7)/(y - 1))#

This means when **range** is:

#color(blue)((-oo, 1) uu (1, oo))#

You can see that this is the case in the graph itself:

graph{(x + 7)/(x - 5) [-73.3, 74.9, -37.07, 36.97]}

What you should notice is the **horizontal asymptote** at **vertical asymptote** at

Because the function is trying to reach an undefined value at those points (*cannot be crossed*, only ascended or descended from either side.