• Interpretation I:
Find f and g, so that
4
(f o g)(x) = —————
x² + 9
Well, there is more than one possibility.
4
For instance, It can be: f(x) = —— and g(x) = x² + 9,
x
and then you have
(f o g)(x) = f[ g(x) ]
4
(f o g)(x) = ————
g(x)
4
(f o g)(x) = ————— ✔
x² + 9
4
Another possibility for that composition: f(x) = ————— and g(x) = x²,
x + 9
and for those, you get
(f o g)(x) = f[ g(x) ]
4
(f o g)(x) = ———————
[ g(x) ]² + 9
4
(f o g)(x) = ————— ✔
x² + 9
As you can see above, there are many ways to find f and g, so the composition of those is (f o g)(x) = 4/(x² + 9).
—————
• Interpretation II:
Find f and g, so that
4
(f o g)(x) = —— + 9
x²
4
It can be: f(x) = x + 9 and g(x) = ——
x²
and then you have
(f o g)(x) = f[ g(x) ]
(f o g)(x) = g(x) + 9
4
(f o g)(x) = —— + 9
x²
2
or it could be also: f(x) = x² + 9 and g(x) = ——
x
and you have again
(f o g)(x) = f[ g(x) ]
(f o g)(x) = [ g(x) ]² + 9
(f o g)(x) = [ 2/x ]² + 9
(f o g)(x) = (2²/x²) + 9
4
(f o g)(x) = —— + 9 ✔
x²
As you can see above, there are many ways to find f and g, so the composition of those is (f o g)(x) = (4/x²) + 9.
I hope this helps. =)
Tags: composite functions rational quadratic linear function algebra