I DON'T HAVE THE SOLUTION
I've got this far:
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? f :[2, \infty) \to \mathbb{R}, f(x)=a-x )
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? dom = [2,\infty))
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? ran = (-\infty,a-2])
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? g:(-\infty,1] \to \mathbb{R}, g(x)= x^2 +a)
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? dom = (-\infty,1] )
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? ran = [a,\infty) )
For
to exist:![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? ran g \subseteq dom f)
For
to exist: ![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?ran f \subseteq dom g)
I've stated the obvious...
Not sure where to go from here.
I did a series of sketches and I came up with this:
which does not look correct at all.
Well, you actually did the right thing in your first couple of steps. You just need confidence in yourself.
f(x) = a - x. Dom = [2, infinity). Range = (-infinity, a - 2]
g(x) = x^2 + a. Dom = (-infinity, 1], Range = [a, infinity)
For f(g(x)) to exist, range of g(x) is a subset or equal to the domain of f(x). Hence, [a,infinity) must be a smaller set than [2,infinity). How is that possible? a must be greater than or equal to 2! In mathematical notation,
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? a \geq 2 )
.
Similarly, for g(f(x)) to exist, the range of f(x) must be a "smaller set" than the domain of g(x). Hence, (-infinity, a - 2] is 'smaller' as a group than (-infinity, 1]. How is that possible? a - 2 must be less than or equal to 1! In maths notation,
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? a - 2 \leq 1 )
. Or, in other words
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi?a\leq 3 )
So now you just find when both of them hold true, i.e. their 'intersection', which will be when
![](https://archive.atarnotes.com/cgi-bin/mathtex.cgi? 2 \leq a \leq 3 )
.
Hope I helped.