Elementary questions about functions, notation, properties, and operations such as function composition.

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0
votes
3answers
38 views

onto but not one-to-one on set of Natural Numbers

Let $\mathbb{N} = \{0, 1, 2, 3, ...\}$. Is there a function from $\mathbb{N}$ to $\mathbb{N}$ which is an onto function but not one-to-one function? I have tried it but could not find any such ...
3
votes
0answers
38 views

System of equations for operations

Given a system with multiple equations, where we know the values and the result, but not the operations between the values: \begin{cases} 3 ⊕ 5 ⊙ 2 = 13 \\ 7 ⊕ 2 ⊙ 4 = 10 \\ 4 ⊕ 3 ⊙ 3 = 9 \end{cases} ...
7
votes
4answers
512 views

Negation of injectivity

I'm having some problems understanding the negation of injectivity. Take the function $f: \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = x^2$. The formal definition of injectivity is $f(a)=f(b) ...
0
votes
1answer
40 views

price and quantity after taxation

Given that demand for a good X is equal to $q_D=393-2p$ and market supply is $q_S=p/4-12$. Find equilibrium price and quantity, consumer and producer surplus and draw a diagram illustrating the ...
-1
votes
2answers
43 views

How is this an Even function? [closed]

How is this, $$f(x) = \left\{\begin{array}{cc} |x + 1|, \quad &-2 \le x \le 0 \\ |x - 1|, \quad& 0 < x \le 2 \end{array}\right.$$ an even function?
2
votes
2answers
32 views

Find the inverse of the function $r(x)=1-2f(3-4x)$ in terms of $f^{-1}$

I have absolutely no idea to inverse functions containing different functions. Apparently this is a one-to-one function with inverse $f^{-1}$ and I'm asked to calculate the inverses of the given ...
7
votes
3answers
123 views

Existence of function satisfying a certain limit

Does there exist a real function $f$ such that $\lim_{x\to+\infty}f(x)=+\infty$ and such that $$\lim_{x\to+\infty}\frac{f(x+1)f(x)}{x(f(x+1)-f(x))}=0?$$ I tried powers, logarithms, exponentials in all ...
1
vote
1answer
41 views

Find local maximum or minimum in 2 variable function

So, I encountered a question (don't worry it's not H.W.) where I have a function with two variables, and I need to find local maximum / minimum points if exists. (More precisely, it is a utility ...
0
votes
2answers
40 views

Definition of non-injective function

Let $f\in F^{E}$ so to show that $f$ is non injective it's suffice to find distinct elements in $E$ with equal images. $$f \text{ is non-injective } \iff \exists (x,y)\in E^{2} \text{ s.t } x\neq y ...
-1
votes
2answers
48 views

How to proof card p(a) = card p(b)? [closed]

I'm stuck on how to demonstrate that card (p(a)) = card (p(b)) but i meed to doo that without using that card (p(a)) = 2^a PS: i already have card(a) = card(b)
2
votes
0answers
74 views

Integer factorization: How to explain those numbers?

I am getting a non primary number $d$ with two nontrivial factors and I am trying to find what they are. Basically I am trying to solve the Integer Factorization Problem in special case. I want to ...
3
votes
2answers
201 views

Sufficient conditions for bounded function?

Consider a function $f:[a,b]\subset\mathbb{R}\rightarrow\mathbb{R}$. Does the fact that the domain of $f$ is a compact set of the real line imply that $f$ is bounded on $[a,b]$? In the negative case, ...
0
votes
3answers
43 views

Quick Clarification: Definition of Bijective Function

I am very familiar with the concepts of bijective, surjective and injective maps but I am interested in improvising the definition of bijection in a way I have not seen done before. To be clear I will ...
0
votes
2answers
56 views

Is $f(x)=\frac{e^x}{1+e^x}$ always monotonically increasing?

Is the function $f(x) = \frac{e^x}{1+e^x}$ always monotonically increasing? As it is bound to $(-1,1)$, I am confused because $f(x)$ reaches a constant value when $x$ approaches infinity. So it ...
1
vote
1answer
37 views

Find $A$ and $B$ Under certain conditions

let $E$ be a finite set with $|E|=n\in \mathbb{N}$ and $$ \begin{array}{ccccc} f_{(A,B)}:& & E & \longrightarrow & \{0,1,2\} \\ & & x & \mapsto & ...
1
vote
1answer
30 views

If $-1\le \alpha<0 $ , show that the function $f(x)=(x+4)^\alpha - 3 (x+6)^\alpha +x(x+3)^\alpha -x(x+5)^\alpha $ is strictly increasing

If $-1\le \alpha<0 $ , show that the function $f(x)=(x+4)^\alpha - 3 (x+6)^\alpha +x(x+3)^\alpha -x(x+5)^\alpha $ is strictly increasing for $x \ge 0 $ . I tried this in many ways using the ...
0
votes
0answers
24 views

How do I find $r(x,x_0)$ of function?

$$r(x_0, x) := \frac{f(x) − f(x_0)}{x − x_0} − f′(x_0)$$ We're given $$f(x)=x^3-4x-2\\ x_0=2$$ I have found $f′(x)=3x^2-4$ and $f′(x_0)=8$. But still I can't get $r(x_0, x)$. If someone could help ...
0
votes
1answer
36 views

How to write a function that will roughly return numbers in this pattern? [closed]

I feel ashamed but I am willing to learn. How do I write a function to get this results for given x? I need a quadratic function right? for x = 0, return 1 for x = 20, return 1.5 for x = 30, ...
0
votes
2answers
84 views

Is there an analytic function with $f(1/n)=a_n \text{ where } a_n \in \{1/2,0,1/4,0,1/6 \ldots\}$

Is there an analytic function such that $$f(1/n)=a_n \text{ where } a_n \in \{1/2,0,1/4,0,1/6 \ldots\}\text?$$ I could not solve this problem. Perhaps I am missing some lemma not shown in class. ...
0
votes
1answer
18 views

Public goods - 2 people buying something

Bob and Ray are thinking of buying a sofa. Bob's utility function is $U_B(S,M_B)=(1+S)M_B$ and Ray's utility function is $U_R(S,M_R)=(3+S)M_R$ where $S=0$ where S=0 if they do not get the sofa and S=1 ...
0
votes
2answers
59 views

Does function: $ f: \mathbb{Z} \to \mathbb{Z} $ exist for these statements

Does function: $ f: \mathbb{Z} \to \mathbb{Z} $ exist, such that this statement is true: $$(\forall{x} \in \mathbb{Z}:f(x) \geq 2)\Rightarrow(\forall{x}\in \mathbb{Z}:f(x)<10)$$ and this statement ...
0
votes
1answer
23 views

Weierstrass theorem and compactness

On my book the statement of Weierstrass theorem is: If $f$ is a continuous function $f:A\subseteq X\rightarrow \mathbb{R}$ defined on a compact set $C$, where $A$ is the domain of $f$ and $X$ is a ...
0
votes
5answers
56 views

What is the Inverse function of $y = 10^{-x}$? Steps are appreciated.

What is the inverse of $y = 10^{-x}$? These are my steps for the problem. Step 1 $y = 10^{-x}$. Step 2 $x = 10^{-y}$ by inverse substitution. Step 3 $10^y(x) = 1$. Step 4 $10^y = ...
-1
votes
2answers
44 views

What is a good book for learning math from the very basic? [closed]

I am an undergraduate student. Lately i realized that i donot have any basic on mathematics, not even elementary level!!can you recommend some books for learning math from the ground up-- from rather ...
8
votes
2answers
213 views

Extending functions from integers to reals in a “nice” way.

For every function $f$ from $\mathbb{Z}$ to $\mathbb{R}$, can we find a function $g$ from $\mathbb{R}$ to $\mathbb{R}$ that is infinitely differentiable and agrees with $f$ on $\mathbb{Z}$? ...
0
votes
1answer
47 views

X and Y intercepts

I have a function: $$y =\begin{cases} 3x, &\text{if } x ≠ 0; \\ 4, & \text{if } x = 0. \end{cases}$$ As I understand, function doesn't exist at (0;0). So, what ...
3
votes
1answer
40 views

Find the minimum roots of $f'(x)\cdot f'''(x)+(f''(x))^2 =0$ given certain conditions on $f(x)$.

Problem: Let $f(x)$ be a thrice differentiable function satisfying: $$|f(x) - f(4-x)| + |f(4-x)-f(4+x)| = 0, \forall x \in R$$ If $f'(1)=0$, then find the minimum number of roots of $f'(x)\cdot ...
1
vote
2answers
34 views

Function with one level set - circle.

I am looking for a function with only one circle level set. In other words : I am looking for function $f : R^n\rightarrow R$ and a set (level set) $\{q\in\ R^n :f(q)=p \}$ when $p = z$(some ...
0
votes
1answer
56 views

Partial differential equality

Let $f \colon \mathbb{R}^2 \setminus \{0,0\} \to$ $\mathbb{R}$ be a smooth function such that $$ x\frac {\partial f(x,y)}{\partial y} + y\frac {\partial f(x,y)}{\partial x} =f(x,y)$$ ...
4
votes
2answers
38 views

Prove that if $f(x)=a/(x+b)$ then $f((x_1+x_2)/2)\le(f(x_1)+f(x_2))/2$

This exercise : If $f(x)=a/(x+b)$ then : $$ f((x_1+x_2)/2)\le(f(x_1)+f(x_2))/2$$ was in my math olympiad today (for 16 years olds). I proved this by saying this is true due to Jensen's ...
-2
votes
1answer
52 views

Can someone explain to me how to find zeros of a function? $10x^2+20x+19x+97^1$

I got this function right here and my teacher wants me to find all real number zeros $$10x^2+20x+19x+97^1.$$ I looked up this video on how to find it and they were using the $P/Q$ and I found ...
3
votes
1answer
63 views

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying $f(x+y^3)=f(x)+f(y)^3$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function satisfying $f(x+y^3)=f(x)+f(y)^3$ for all $x,y\in\mathbb{R}$. If $f'(0)\ge0$, find $f(10)$. If $x=y=0$, $f(0+0)=f(0)+f(0)$. So $f(0)=0$. If ...
3
votes
4answers
79 views

Is $x^2$ $+$ $y^2$ = $25$ a function?

General Question: Is $x^2$ $+$ $y^2$ = $25$ a function? If I input 2 different x's and it resulted in only 1 y value for each. This is definitely a function right?
2
votes
0answers
55 views

compact set in a space of functions continuous in $\mathbb{R}$

As it is known the set $\mathcal{B}=\{ f:\mathbb{R}\rightarrow \mathbb{R} : f \mbox{ is continuous}\}$ it is not a metric space with the metric $d(f,g)=sup_{x\in \mathbb{R}}\| f(x)-g(x)\|$ it can ...
5
votes
1answer
38 views

Functions where the pre-image of convex sets is convex

For functions $f:\mathbb R\to\mathbb R$, I've noticed an interesting property: $f$ is monotonous exactly if the pre-images of convex sets are convex. Now the latter condition can of course be defined ...
0
votes
0answers
28 views

Modulus of a 32bit number using 16 bit numbers.

I am trying to calculate the modulo of 2 32bit numbers using 16bit numbers only. a mod b = x if 'a' is greater than a 16 bit I can rewrite the equation as : ...
4
votes
1answer
58 views

Finding $f\in C( \mathbb R)$ such that for some integer $n>1$, $f^n(x)=x,\,\forall x \in \mathbb R$

Let $f:\mathbb R \to \mathbb R$ be a continuous function such that for some integer $n>1$, $f^n(x)=x,\,\forall x \in \mathbb R$; then is it true that either $f(x)=x,\,\forall x \in \mathbb R$ or ...
0
votes
1answer
30 views

biholomorphic functions=bijective+holomorphic?

I have a question on biholomorphic functions. I saw the following theorem in lecture: Let $U,V\subseteq \mathbb{C}$ open subsetes and $f:U\to V$ holomorphic, bijective and $f'(z)\neq 0$ for all $z\in ...
0
votes
2answers
45 views

Method for finding the norm of an operator by constructing a sequence.

I asked a similar question to this one before, and while I am interested to get an answer to the following problems, I am primarily concerned with the actual method for constructing the sequence in ...
1
vote
0answers
31 views

Beginner deravative of peice wise question and connection with differentiation and continouity

Hi I am wondering if someone can help to explain to me the following; Say we have $$f(x)= \begin{cases} x^{2}, &\text{ if $x \ge 0$ } \\ 0, &\text{if $x \lt 0$} \\ \end{cases}$$ and we want ...
0
votes
3answers
88 views

Finding the function that would describe this:

I'm not going to go into detail why I am interested in the next iteration of these functions, but here they are: 1: 6/(x+1) 2: 8/(2^x) 3: 10/(?) The question is, which one is next? I will say that ...
1
vote
1answer
28 views

Difference between function and equation

What is the precise difference between function and equation ? In which case will it be wrong if used( common mistakes )? Also will the Venn diagram overlap if I were to draw one ? Any help and ...
2
votes
0answers
37 views

Find all such $a$ that $x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|$ has at least one root.

In the equation, $a$ is a parameter and $x$ is a variable: $$x+2\lvert x-3 \rvert = 7\lvert x-a \rvert + 3 \lvert x-a-4|.$$ I want to find all values of $a$ that make the equation have at least one ...
1
vote
0answers
35 views

Proper notation to indicate intervals in which a function is increasing

I will ask my question with an example. Define $f:[0, 2\pi] \rightarrow \mathbb{R}$ such that $f(x) = \sin(x)$. This function is increasing in the intervals $(0, \frac{\pi}{2})$ and $(\frac{3\pi}{2}, ...
0
votes
0answers
28 views

What does notation $C^{\beta}[0,1]$ mean?

What does the notation $C^{\beta}[0,1]$ for $\beta \in (0,1]$ mean? I know $C[0,1]$ is the space of all continuous functions on the interval $[0,1]$, but what about $C^{\beta}[0,1]$? Usually ...
0
votes
1answer
13 views

Find area of triangle which sides is limited by two functions and the x axis

I'm studying for my math exam and I'm stuck on the following question "A triangle is limited by the x axis and the two functions $y=kx$ och $y=\frac{1}{k}x+k$ where k > 1. Determine the smalest ...
-2
votes
1answer
22 views

count the numer of partial function between two sets with bijection [duplicate]

A and B two sets. |A|=k,|B|=n. prove: the number of partial function from A to B is (n+1)^k. proof: x-object not in B. B*=BUx. The total function (not partial) between A and B* is (n+1)^k. Now I need ...
0
votes
2answers
31 views

Find the number of local extrema of that function without calculus.

I need to find the number of local extrema of that function without derivate or using calculus. I know that in $x = 1$ and $x = 3$ $f(x) = 0$ ... in which way I can affirm that this function has at ...
1
vote
4answers
30 views

Find range of composite trigonometric function

Find the range of the function: $$f(x)= \cos^2x-\cos x $$ Answer is: $ [-1/4 ,2]$ I've factorized $\cos x$ and thus got min and max values equal to $[0,2]$ using inequalities. I know it's $-1/4$ ...
1
vote
2answers
29 views

Inverse Function That Includes Fraction [closed]

Nice inverse function I am struggling on, totally forgotten how to move the fraction over: $$g(x)= \frac 1 {x-2}+5 \qquad (x>2)$$ Find the inverse function $g^{-1}$ specifying the rule domain and ...