# How do you find the domain and range of #f(x) = sqrt(4+x) / (1-x)#?

##### 1 Answer

Domain:

Range:

#### Explanation:

The **domain** of the function represents all the values that **defined**.

Right from the start, you should be able to say that the domain of the function **cannot** include *undefined*.

#1 - x != 0 implies x != 1#

Moreover, notice that the function contains the square root of an expression that depends on the value of *real numbers*, you cannot take the square root of a *negative number*.

This implies that you need

#4 + x >= 0 implies x >= - 4#

Therefore, you can say that the domain of the function will be

#x in [-4, 1) uu (1, +oo)#

THis tells you that the function is defined for any value of

The **range** of the function tells you all the possible values that

In this case, the square root of a positive number will produce a positive number, which means that **regardless** what value of

#sqrt(4 -x ) >= 0 #

Now, for any value of

#{( sqrt(4 + x) >= 0), (1 - x > 0) :} implies f(x) >= 0#

and for any value of

#{( sqrt(4 + x) > 0), (1 - x < 0) :} implies f(x) < 0#

This means that the range of the function is

#(-oo, 0] uu (0, +oo) = (- oo, +oo)#

graph{sqrt(4+x)/(1-x) [-10, 10, -5, 5]}