SAT1000 - P879

Algebra Level pending

Let AA be the set of all functions whose range is R\mathbb R, BB is the set of all functions ϕ(x)\phi(x) which has the following properties:

  • For ϕ(x)\phi(x), let R0R_0 denote the range of ϕ(x)\phi(x), M(0,+),R0[M,M]\exists M \in (0,+\infty), R_0 \subseteq [-M,M].

It's easy to prove that for ϕ1(x)=x3\phi_1(x)=x^3, ϕ2(x)=sinx\phi_2(x)=\sin x, ϕ1(x)A\phi_1(x) \in A, ϕ2(x)B\phi_2(x) \in B.

Here are the following statements:

  1. Let DD be the domain of f(x)f(x), then the necessary and sufficient condition for f(x)Af(x) \in A is: bR,aD,f(a)=b\forall b \in \mathbb R, \exists a \in D, f(a)=b.

  2. The necessary and sufficient condition for f(x)Bf(x) \in B is f(x)f(x) has the maximum and minimum value.

  3. If f(x),g(x)f(x), g(x) have the same domain, then if f(x)A,g(x)Bf(x) \in A, g(x) \in B, then f(x)+g(x)Bf(x)+g(x) \notin B.

  4. If f(x)=aln(x+2)+xx2+1 (x>2,aR)f(x)=a\ln(x+2)+\dfrac{x}{x^2+1}\ (x>-2, a \in \mathbb R) has the maximum value, then f(x)Bf(x) \in B.

Which statements are true?

How to submit:

Let p1,p2,,pnp_1, p_2,\cdots,p_n be the boolean value of statement 1,2,,n1,2,\cdots,n, if statement kk is true, pk=1p_k=1, else pk=0p_k=0.

Then submit k=1npk2k1\displaystyle \sum_{k=1}^n p_k \cdot 2^{k-1}.

Have a look at my problem set: SAT 1000 problems


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