This question is all about domain.
demands that is in the domain of .
Whereas, demands that is in the domain of .
Can you see that works BUT does not?
Now work on it.
Hope I've posted this question in the right place- I think I did.. But anyway.
Let a be a positive number, let f : [2,∞) → R, f (x) = a − x and let g: (−∞, 1] → R,
g(x) = x^2 + a. Find all values of a for which f ◦ g and g ◦ f both exist.
I've no clue how to do this could someone show me?
No, he is not suggesting that you "try numbers at random" but he is giving two examples so you can see what is happening. We are given that the domain of f is not "all real numbers" but "all real numbers greater than or equal to 2". In order that f(g(x)) exist, we must have . That says where, remember, x must be less than or equal to 1. What must a be so that ?
In order that g(f(x)) exist, f(x) must be less than or equal to 1. That is, . But x must be greater than or equal to 2. What must a be so that ?