# How do you find the domain and range of #g(x)=sqrt((x+3)/(x-2))#?

##### 1 Answer

Range:

Domain:

#### Explanation:

In order to find the **domain** of this function, you need to find all the values that **defined**.

Right from the start, you know that the denominator of the fraction **cannot** be equal to *undefined*.

#x - 2 != 0 implies x != 2#

Now, you know that when working with *real numbers*, you can only take the square root of a **positive number**.

This implies that you must have

#(x+3)/(x-2) >= 0#

You know that when

#(-3 + 3)/(-3 - 2) = 0/(-5) >= 0 #

so you can say that

Now, in order to have

#(x+3)/(x-2) > 0#

you must look at two possible situations

#x + 3 >0" " ul(and) " " x -2 > 0# In this case, you must have

#x + 3 > 0 implies x >= -3# and

#x - 2 > 0 implies x > 2# This implies that the solution interval will be

#(-3, + oo) nn (2, + oo) = (2, + oo)# This tells you that any value of

#x# that is greater than#2# will satisfy the inequality#(x+3)/(x-2) > 0# .

#x + 3 <0" "ul(and)" " x - 2 < 0# In this case, you must have

#x + 3 < 0 implies x < -3# and

#x - 2 < 0 implies x < 2# This implies that the solution interval will be

#(- oo, -3) nn (-oo, 2) = (-oo, -3)# This tells you that any value of

#x# that is less than#-3# will also satisfy the inequality#(x + 3)/(x - 2) > 0# .

Therefore, the domain of the function will be--remember that

#"domain: " color(darkgreen)(ul(color(black)(x in (-oo, -3] uu (2, oo))))#

This tells you that any value of **or** greater than

#(x+3)/(x-2) >= 0#

Now, to find the **range** of the function, you must determine the values that

Since you're working with real numbers, you can say that taking the square root of a positive number will always return a **positive number**.

#g(x) >= 0 #

You know that when

#g(-3) = sqrt( (-3 + 3)/(-3 - 2)) = 0#

Now, it's important to realize that the range of the function will **not** include **never** have

#color(red)(cancel(color(black)(x))) + 3 = color(red)(cancel(color(black)(x))) -2#

#3 != - 2#

This means that you don't have a value of

Therefore, the range of the function will be

#"range: " color(darkgreen)(ul(color(black)(g(x) in [0, 1) uu (1, + oo))))#

graph{sqrt( (x+3)/(x-2)) [-16.02, 16.01, -8.01, 8]}