# What is the domain and range of #f(x) = (x+9)/(x-3)#?

##### 2 Answers

Domain:

Range:

#### Explanation:

**Domain**

The domain of a function is the set of points in which the function is defined. With numeric function, as you probably know, some operations are not allowed - namely division by

In your case, you have no logarithms nor roots, so you only have to worry about the denominator. When imposing

**Range**

The range is an interval whose extrema are the lowest and highest possible values reached by the function. In this case, we already notices that our function has a point of non-definition, which leads to a vertical asymptote. When approaching vertical asymptotes, functions diverge towards

In fact, if

By the same logic,

Since the function approaches both

#### Explanation:

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.

#"solve "x-3=0rArrx=3larrcolor(red)"excluded value"#

#"domain "x in(-oo,3)uu(3,oo)#

#"let "y=(x+9)/(x-3)#

#"rearrange making x the subject"#

#y(x-3)=x+9#

#xy-3y=x+9#

#xy-x=9+3y#

#x(y-1)=9+3y#

#x=(9+3y)/(y-1)#

#"solve "y-1=0rArry=1larrcolor(red)"excluded value"#

#"range "y in(-oo,1)uu(1,oo)#

graph{(x+9)/(x-3) [-10, 10, -5, 5]}