Let t = x 2+1

The monotone interval and monotonicity of quadratic function can be found.

After substitution, there is a function f (t) = 5+2t-t 2.

Finding its monotone interval and monotonicity on the interval is also a quadratic function.

These two functions form a composite function, that is, f[g(x)].

In this problem, the inner function is t = x 2+ 1 and the outer function is y = 5+2t-t 2.

Note that the monotone interval should take its common part. For example, the monotone interval of t = x 2+1 is (negative infinity, 0, and the monotone interval of y = 5+2t-t 2 is (negative infinity,1), and the conclusion should be (negative infinity, 0).

Then judge according to the same increase and different decrease.

I figured it out, I don't know if it's right, but it's also a subtraction function on (negative infinity, 0 is a subtraction function on 1, positive infinity).