1. ## Composite Functions Question

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?

2. This question is all about domain.
$g\circ f$ demands that $f(t)$ is in the domain of $g$.
Whereas, $f\circ g$ demands that $g(t)$ is in the domain of $f$.
Can you see that $a=3$ works BUT $a=0$ does not?

Now work on it.

3. Yes.. but where do you get the figure 3 from? You don't just randomly try out numbers do you ?

4. 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 $g(x)\ge 2$. That says $x^2+ a\ge 2$ where, remember, x must be less than or equal to 1. What must a be so that $g(1)= 1+ a\ge 2$?

In order that g(f(x)) exist, f(x) must be less than or equal to 1. That is, $f(x)= a- x\le 1$. But x must be greater than or equal to 2. What must a be so that $f(2)= a- 2\le 1$?

5. Oh now it makes sense. So a = 2 or 3.
Thanks guys

6. It also could be 2.5.