Bài tập Hàm số


1. Find all functions f from the set \mathbb{R} of real numbers into \mathbb{R} which satisfy for all x, y, z \in \mathbb{R} the identity

f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).

2. Consider the function f: \mathbb{N}_0\to\mathbb{N}_0, where \mathbb{N}_0 is the set of all non-negative integers, defined by the following conditions

(i)f(0)=0;

(ii)f(2n)=2f(n) and

(iii)f(2n+1)=n+2f(n) for all n\geq 0.

(a)Determine the three sets L=\{ n | f(n) < f(n+1) \}, E=\{n | f(n)=f(n+1)\}, and G=\{n | f(n) > f(n+1)\}.

(b)For each k\geq 0, find a formula for a_k=\max\{f(n) : 0 \leq n \leq 2^k\} in terms of k.

3. Let n be a positive integer. Find the largest nonnegative real number f(n) (depending on n) with the following property: whenever a_1,a_2,...,a_n are real numbers such that a_1+a_2+\cdots +a_n is an integer, there exists some i such that  \left|a_i-\dfrac{1}{2}\right|\ge f(n).

4. Let {\bf R} denote the set of all real numbers. Find all functions f from {\bf R} to {\bf R} satisfying

(i)There are  only finitely many s in {\bf R} such that f(s)=0,

And

(ii)f(x^4+y)=x^3f(x)+f(f(y)) for all x,y in {\bf R}.

5. Find all a\in\mathbb{R} for which there exists a non-constant function f: (0,1]\rightarrow\mathbb{R} such that a+f(x+y-xy)+f(x)f(y)\leq f(x)+f(y)

for all x,y\in(0,1].

6. Consider function f: \mathbb{R}\to\mathbb{R} which satisfies the conditions for any mutually distinct real numbers a,b,c,d satisfying \dfrac{a-b}{b-c}+\dfrac{a-d}{d-c}=0, f(a),f(b),f(c),f(d) are mutully different and

\dfrac{f(a)-f(b)}{f(b)-f(c)}+\dfrac{f(a)-f(d)}{f(d)-f(c)}=0. Prove that function f is linear.

7. Find all complex polynomial P(x) such that for any three integers a,b,c satisfying a+b+c\not=0, \dfrac{P(a)+P(b)+P(c)}{a+b+c} is an integer.

8. Find all functions f:\mathbb{Q}^{+}\to\mathbb{Q}^{+} such that

f(x)+f(y)+2xyf(xy)=\dfrac{f(xy)}{f(x+y)}\forall x,y\in\mathbb{Q}^+.

9. Let \alpha be given positive real number, find all the functions f:\mathbb{N}^{+}\to\mathbb{R} such that f(k + m) = f(k) + f(m) holds for any positive integers k, m satisfying \alpha m \leq k \leq (\alpha + 1)m.

10. Given non-zero reals a, b, find all functions f: \mathbb{R} \to \mathbb{R}, such that for every x, y \in \mathbb{R}, y \neq 0, f(2x)=af(x)+bx and f(x)f(y)=f(xy)+f\left(\dfrac{x}{y}\right).

11. Prove that for all integers a > 1 and b > 1 there exists a function f from the positive integers to the positive integers such that f(a\cdot f(n))=b\cdot n for all n positive integer.

12. Find all functions f:\mathbb{R}\to\mathbb{R} such that f(xf(y)+f(x)) = 2f(x)+xy for every reals x,y.

13. Let f(x) be a real-valued function defined on the positive reals such that

(1)If x < y, then f(x) < f(y),

(2)f\left(\dfrac{2xy}{x+y}\right) \geq\dfrac{f(x) + f(y)}{2} for all x,y>0.

Show that f(x) < 0 for some value of x.

14. Define f on the positive integers by f(n) = k^2 + k + 1, where n=2^k(2l+1) for some k,l nonnegative integers. Find the smallest $n$ such that

f(1) + f(2) + ... + f(n) \geq 123456.

15. Find all functions f:\mathbb{N}\to\mathbb{N} satisfying, for all x\in\mathbb{N}, f(2f(x)) = x + 1998.

16.  a) Show that there are no functions f,g: \mathbb{R}\to\mathbb{R} such that g(f(x)) = x^3 and f(g(x)) = x^2 for all x\in\mathbb{R}.

b)Let S be the set of all real numbers greater than 1. Show that there are functions f,g:S\to S satsfying the condition above.

17. Let f(x)= x^2-C where C is a rational constant. Show that exists only finitely many rationals x such that \{x,f(x),f(f(x)),\cdots\} is finite.

18. Find all real-valued functions on the positive integers such that f(x + 1019) = f(x) for all x, and f(xy) = f(x)f(y)for all x,y.

19. Find at least one function f:\mathbb{R}\to\mathbb{R} such that f(0)=0 and f(2x+1) = 3f(x) + 5 for any real x.

20. Let f(x)=\dfrac{ax+b}{cx+d}, F_n(x)=f(f(f\cdots (f(x))\cdots)) (with n\ f's). Suppose that f(0)\not =0, f(f(0))\not =0, and for some n we have F_n(0)=0, show that F_n(x)=x (for any valid x).

21. Find all functions f: \mathbb{Z}\rightarrow\mathbb{Z} such that for all x,y \in \mathbb{Z} f(x-y+f(y))=f(x)+f(y).

22. Find all functions f: \mathbb{R} \to \mathbb{R} such that \forall x,y,z\in\mathbb{R} we have: If x^3+f(y) \cdot x+f(z)=0, then f(x)^3+y \cdot f(x)+z=0.

23. Let S\subseteq\mathbb{R} be a set of real numbers. We say that a pair (f, g) of functions from S into S is a Spanish Couple on S, if they satisfy the following conditions

(i) Both functions are strictly increasing, i.e. f(x) < f(y) and g(x) < g(y) for all x, y\in S with x < y;

(ii) The inequality f\left(g\left(g\left(x\right)\right)\right) < g\left(f\left(x\right)\right) holds for all x\in S.

Decide whether there exists a Spanish Couple

a)On the set S=\mathbb{N} of positive integers;

b)On the set S=\{a-\dfrac{1}{b}: a, b\in\mathbb{N}\}.

24. For every n\in\mathbb{N} let d(n) denote the number of (positive) divisors of n. Find all functions f:\mathbb{N}\to\mathbb{N} with the following properties:

a)d\left(f(x)\right)=x for all x\in\mathbb{N}, and

b)f(xy) divides (x-1)y^{xy-1}f(x) for all x, y\in\mathbb{N}.

25. Consider those functions f:\mathbb{N}\to\mathbb{N} which satisfy the condition f(m+n) \geq f(m)+f(f(n))-1 for all m,n\in\mathbb{N}. Find all possible values of f(2007).

26. Find all surjective functions f:\mathbb{N}\to\mathbb{N} such that for every m,n\in\mathbb{N} and every prime p, the number f(m+n) is divisible by p if and only if f(m)+f(n) is divisible by p.

27. Find all real polynomials f such that 2yf(x+y)+(x-y)(f(x)+f(y)) \geq 0\forall x,y\in\mathbb{R}.

28. Determine all functions f:\mathbb{R}\to\mathbb{R} with x,y\in\mathbb{R} such that f(x-f(y))=f(x+y)+f(y).

29. Show that for positive integer n, and for x\not =0,

\left(x^{n-1}\sin\dfrac{1}{x}\right)^{(n)}=\dfrac{(-1)^n}{x^{n+1}}\sin\left(\dfrac{1}{x}+\dfrac{n\pi}{2}\right).

30. Find all f:\mathbb{R}\to\mathbb{R} such that

f(xy+f(x))=xf(y)+f(x)

for every pair of real numbers x,y.

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