# What is the domain and range of #y=sqrt((x+5) (x-5))#?

##### 1 Answer

Domain:

Range:

#### Explanation:

The **domain** of the function will include all the values that **defined**.

In this case, the fact that you're dealing with a square root tells you that the expression that's under the square root sign must be **positive**. That is the case because when working with *real numbers*, you can only take the square root of a **positive number**.

This means that you must have

#(x + 5)(x - 5) >=0#

Now, you know that for

#(x+5)(x - 5) = 0#

In order to determine the values of

#(x+5)(x-5) > 0#

you need to look at two possible scenarios.

#x+5 > 0 " " ul(and) " " x-5 > 0# In this case, you must have

#x + 5 > 0 implies x > - 5# and

# x - 5 > 0 implies x > 5# The solution interval will be

#(-5, + oo) nn (5, + oo) = (5, + oo)#

#x + 5 < 0 " " ul(and) " " x- 5 < 0# This time, you must have

#x + 5 < 0 implies x < -5# and

# x - 5 < 0 implies x < 5# The solution interval will be

#(-oo, - 5) nn (-oo, 5) = (-oo, - 5)#

You can thus say that the domain of the function will be--**do not** forget that

#"domain: " color(darkgreen)(ul(color(black)(x in (-oo, - 5] uu [5, + oo)#

For the range of the function, you need to find the values that

You know that for real numbers, taking the square root of a positive number will produce a **positive number**, so you can say that

#y >= 0 " "(AA)color(white)(.) x in (-oo, -5] uu [5, + oo)#

Now, you know that when

#y = sqrt((-5 + 5)(-5 - 5)) = 0" " and " " y = sqrt((5 + 5)(5 - 5)) = 0#

Moreover, for every value of

#y >= 0#

This means that the range of the function will be

#"range: " color(darkgreen)(ul(color(black)(y in (-oo"," + oo)))#

graph{sqrt((x+5)(x-5)) [-20, 20, -10, 10]}