Let A be the set of all functions whose range is R, B is the set of all functions ϕ(x) which has the following properties:
- For ϕ(x), let R0 denote the range of ϕ(x), ∃M∈(0,+∞),R0⊆[−M,M].
It's easy to prove that for ϕ1(x)=x3, ϕ2(x)=sinx, ϕ1(x)∈A, ϕ2(x)∈B.
Here are the following statements:
Let D be the domain of f(x), then the necessary and sufficient condition for f(x)∈A is: ∀b∈R,∃a∈D,f(a)=b.
The necessary and sufficient condition for f(x)∈B is f(x) has the maximum and minimum value.
If f(x),g(x) have the same domain, then if f(x)∈A,g(x)∈B, then f(x)+g(x)∈/B.
If f(x)=aln(x+2)+x2+1x (x>−2,a∈R) has the maximum value, then f(x)∈B.
Which statements are true?
How to submit:
Let p1,p2,⋯,pn be the boolean value of statement 1,2,⋯,n, if statement k is true, pk=1, else pk=0.
Then submit k=1∑npk⋅2k−1.
Have a look at my problem set: SAT 1000 problems