Question

Let f(x)=x² - 4 and g(x)=√x -7Find (f °g)(x) and the domain of (f °g)(x)

Answer 1

Let us write our functions:

`f(x)=x^2-4,g(x)=\sqrt[]{x-7}`

Now,

`(f○g)(x)=f(g(x))=f(\sqrt[]{x-7})=(\sqrt[]{x-7})^2-4=(x-7)-4=x-11`

Shorter,

`(f○g)(x)=x-11`

On the other hand, we need to remember that the domain of a composition is the domain of the first function to be composed; in this case, it's g(x). The domain of g(x) consists of all numbers x such that

`x-7\ge0`

for the square root of negative numbers makes no sense. Then,

`x\ge7`

The domain of g(x) consists of all numbers greater than or equal to 7, and the domain of f(g(x)) as well, by what we said above.