Composite Functions Question

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December 4th 2010, 02:44 PM
Klutz

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?
December 4th 2010, 03:59 PM
Plato

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.
December 4th 2010, 06:31 PM
Klutz

Yes.. but where do you get the figure 3 from? You don't just randomly try out numbers do you ?
December 5th 2010, 03:03 AM
HallsofIvy

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 $f(2)= a- 2\le 1$ ?
December 5th 2010, 11:34 AM
Klutz

Oh now it makes sense. So a = 2 or 3.
Thanks guys :)
December 5th 2010, 11:57 AM
Plato

It also could be 2.5.