SAT1000 - P879

Let $A$ be the set of all functions whose range is $\mathbb R$ , $B$ is the set of all functions $\phi(x)$ which has the following properties:

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

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

Here are the following statements:

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

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

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

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

Which statements are true?

How to submit:

Let $p_1, p_2,\cdots,p_n$ be the boolean value of statement $1,2,\cdots,n$ , if statement $k$ is true, $p_k=1$ , else $p_k=0$ .

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

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Have a look at my problem set: SAT 1000 problems