Finding the Domain of Composite Functions With Square Roots

by Salamat Mujeeb
(Sialkot, Pakistan)

Given the following functions:

How do I find the domain of the composite functions fog and gof?

Answer by Pablo:

Whenever we have functions with square roots, the domain will be the set of numbers for which the expression inside the root isn't negative. With that in mind, let's first find the expressions of the composite functions:

For this expression to be well defined we need that the expression inside be positive or zero:

Now we need to work this inequality algebraically to get a more useful condition. First, we add 4 to both sides:

Now we square both sides:

And now we add 2 to both sides:

So, the domain of fog is the set of real numbers greater than or equal to 18. In set notation:

Now, for the function gof we follow the same steps. First, we find the expression:

For this to be well defined we need that:

So, the domain of gof is:

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