# Question #56413

##### 1 Answer

#### Answer:

For part **(a)**: Domain:

For part **(b)**: Domain:

For part **(c)**: Domain:

#### Explanation:

In all three cases, you're dealing with functions that equal the square root of an expression.

Right from the start, you need to take into account the fact that, for real numbers, you **cannot** take the square root of a *negative number*.

This means that you need to check these expressions for any value of *negative*, and exclude these values from the functions' domain.

So, let's start with the first one.

#f(x) = sqrt(-3x-9)#

You need to have

#(-3x - 9)>=0#

#-3x >= 9 implies x <= 9/((-3)) = -3#

The domain of the function will thus include **any value** of

graph{sqrt(-3x-9) [-14.24, 14.24, -7.12, 7.12]}

Do the same for the second one.

#g(x) = sqrt(x^2 + 4)#

Now, because **all real values** of

This means that you don't have any restrictions for the function's domain, which will be

graph{sqrt(x^2 + 4) [-41.1, 41.14, -20.53, 20.56]}

Now for the third one.

#h(x) = sqrt(2x-4) + sqrt(-5x + 20)#

You need the expressions that are under the square roots to be valid at the same time. You will thus have

#2x - 4 >=0#

#2x >= 4 implies x >=2#

and

#-5x + 20 >=0#

#-5x >= - 20 implies x <= 4#

The domain of the function will thus be

graph{sqrt(2x-4) + sqrt(-5x + 20) [-16, 16.05, -8, 8.02]}