# Tagged Questions

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

90 views

### Functional equation: Finding $f(100)$

A polynomial of degree 98 such $f (k)=1/k$ for $k=1,2,3...,98,99$ exists. How to find $f(100)$? What are the possible methods ?
35 views

...
2k views

### How to divide by a matrix

I found a question in an old exam, where the function $\phi(z) := \frac{\exp(z) - 1}{z}$ is given. Now we evaluate $\phi(\mathbf{A})$. But how do I divide by a matrix? I already thought about ...
68 views

### Approximate function as $x$ tends to infinity

I'm looking for a way to approximate the following function $f$ as $x \to \infty$ $$f = \ln \left( 1 + e^{a_1 x} + e^{a_2 x} + A e^{(a_1+a_2) x} \right)$$ where $a_1$, $a_2$ and $A$ are constants. ...
60 views

### Why ordered sequences can be reduced to sets?

I am trying to understand why ordered sequences can be reduced to basic sets. I understand most of the following proof: Sequences can be defined as functions Functions are a special case of ...
37 views

### Drawing squares of functions

If I want to sketch the square of the sinc function, or any function for that matter, is there a neat transformation technique which would allow one not to refer to graphing devices for this task?
54 views

### Find the maximum of a non-linear function with 4 parameters

I'm trying to find the maximum of a function with 4 positive parameters : $$f(x,y,z,t)=$$$$(-2(x+5)^2+200x)+(-2(y+10)^2+200y)+(-2(z+15)^2+200z)+(-2t^2+200t)$$ with $x+y+z+t = 150$ I don't know if ...
113 views

### Pigeonhole principle 3

I need help on this question, I'm lost and really don't know how to proceed: Use the pigeonhole principle to prove that in a round-robin chess tournament (with 18 participants) there will be at least ...
34 views

### Find the maximum of a function with 4 parameters

I'm trying to find the maximum of a function with 4 positive parameters : $$f(x,y,z,t)=(2x+2)+(4y-1)+(3z+4)+(5t+3)$$ with $x+y+z+t = 50$ I don't know if this is feasible and how to proceed. I have ...
37 views

### General formula for a specific problem?

I have a problem which I would like to have a general formula for. Here is the description. There are island aligned by a grid. Every cell contains an island. Every adjacent island is connected by ...
30 views

### Find functions that satisfy this equation.

Give some examples of functions, $F$ and $G$ such that $$x=\sqrt{F(x)+G(x)\sqrt{F(x+n)}}-\sqrt{F(x+n)}.$$ $n$ can be a constant. : with $n\gt{0}$
51 views

### Function with increasing property.

Prove that $\frac{1}{2}(x+2)^{-3/2}-(\frac{1}{2}x+3)(x+3)^{-3/2}$ is increasing function for $x\ge4$. I tried it by taking its first derivative but by first derivative for me its difficult to say it ...
112 views

### Generalization of Cantor Pairing function to triples and n-tuples

Is there a generalization for the Cantor Pairing function to (ordered) triples and ultimately to (ordered) n-tuples? It's however important that the there exists an inverse function: computing z from (...
60 views

### Why is there no inverse function in this case?

There is no inverse function for $R(q) = 40q - 4q^2$ for $0≤q≤10$ But when $0≤q≤5$ there is inverse function. It's something about the function being one-to-one but I don't know why it isn't one-to-...
43 views

### Is this relation symmetric and transitive?

Set A is given as $A = \{1,2,3,4,5,6,7,8,9,10,11,13,14\}$ And is defined as $R = \{(x,y) : 3x = y\}$ The relation that I'm getting is: $R = \{(3,3), (6,6), (9,9), (12,12)\}$ Over here, it is ...
27 views

### If PQis a focal chord, show that the interval RU is parallel to the axis of the parabola.

For part (c) of question thirteen am I only required to find the gradient of RU and prove that is it zero? This is how I have interpreted this question. ANY help on the matter is much appreciated ...
50 views

### Finding the correct slope.

To determine the slope of the graph of this relation do I take the two points as (4,20), (0,0) and then proceed to take 20-0=20 and 4-0=4, to divide 20 by 4 to get the slope of 5m? For the second ...
55 views

### how many matches are played

so the question is the are a total of 16 teams. Each team will play each other once so each team will play 15 matches since they can't play themselves. How many matches in total are played?
110 views

### $f(x)$ is a periodic function. What is its period?

Suppose that $f(x)$ is a periodic function. If we have: $$\forall x :f(x+346)=\frac{1+f(x)}{1-f(x)}$$ What is its minimum period?
1k views

### How to find graph of the sum of two functions

Suppose I know the graphs of two functions $f(x)$ and $g(x)$. How can I find the graph of $h(x)=f(x)+g(x)$? What are the rules to be followed ? P.S. In case my question seems silly,at least provide ...
2k views

### When I was teaching absolute function properties, I suddenly made this question …

I was teaching absolute function properties in a K-12 class. I made this question in my mind. Suppose $f(x)$ is a one-to-one function, and its definition is $f(x)=max\left \{ x,3x\right \}=ax+b|x|+c$...
882 views

### Must all Lebesgue integrable functions really be invertible?

I am studying Lebesgue integration after a course on Riemann integration, and the definition of measurable function is given as follows: $f:{\mathbb R}\rightarrow {\mathbb R}$ is measurable if the ...
51 views

### General Question about number of functions

I am wondering if there is any sort of algorithm , or if not, at least some general approach to the following; Lets say we have two finite sets $$A=\{a_1,a_2,…a_n\}$$ and $$B=\{b_1,b_2,…,b_m\}$$ ...
99 views

### Show that an irrationally periodic function is also a constant function [duplicate]

Let $f:\mathbb R \to \mathbb R$ be a function such that for any irrational number $r$, and any real number $x$ we have $f(x)=f(x+r)$. Show that $f$ is a constant function.
36 views

### Characteristics of a logarithmic function [closed]

For my math course I have been given a practice question which I am not sure how to go about. For the function: $$g(x) = 2 \log_{10} (x +1)$$ i) State the domain and the equation of the asymptote. ...
95 views

### What is meaning of this question and how to solve it?

I am stuck with understanding the meaning of the question, which states: Show that $\cos(n\theta)=f_n(\cos\theta)$ for polynomials $f_n(x)$ satisfying $$f_{n+1}(x)=2xf_n(x)-f_{n-1}(x) \tag{1}$$ ...
69 views

### How to Prove It Exercise 7.2.5

Prove that ${}^{\mathbb{Z}^+} \mathcal{P}(\mathbb{Z}^+) \sim \mathcal{P}(\mathbb{Z}^+)$ where ${}^A B$ means the set of all functions $f:A \rightarrow B$ and $\mathcal{P}(A)$ is the power set of $A$. ...
52 views

290 views

### $f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h$, find $f(7)$.

Problem : $f(x) =ax^6 +bx^5+cx^4+dx^3+ex^2+gx+h$ Given that : $f(1)= 1, f(2) =2 , f(3) = 3, f(4) =4, f(5)=5, f(6) =6$ find $f(7) =?$ My approach: We can put the values of $f(1) = 1$ in the ...
72 views

### Is it always possible to converge from an integer to another integer? [closed]

Let's say I'm given a fixed integer, I. I'd like to know if it is always possible to find a function, that starting from any random integer J will converge to or oscillate reasonably close (let's say ...
53 views

### Why is this proof for an arbitrary function constrained to a constant one?

Sorry if this seems trivial, I'm having some difficulty understanding a proof. I'm doing exercise 5.1.14 of Velleman's How to Prove It and a solution posted in this question, including the comments, ...
67 views

### Prove that equality holds only if $f$ is one-to-one.

I am just looking for a hint. Not a solution as I am just trying to solve these for fun. Let $f:A \rightarrow B$ with $A_0 \subset A$ and $B_0 \subset B$. Show that $$A_0 \subset f^{-1}(f(A_0))$$ ...
115 views

### What is the precise difference between functions and operators?

I have heard affirmatively that all operators are functions, but not all functions are operators. But at the same time I have heard that functions map numbers to numbers, whereas operators map ...
59 views

### Prove that the unique zeros of $f(x,y)=a x +(1-a)y+xy$ when $x,y\in[0,1]$, is $x=y=0$.

Prove that the unique zeros of the two-variables function: $$f(x,y)=a x +(1-a)y+xy$$ when $x,y\in[0,1]$, is $x=y=0$. Here, $a$ is a parameter between 0 and 1. I have no idea where to start. Any ...
46 views

### Can a limit of multivariable function can be taken componentwise?

Is there a theorem saying that $\lim_{(x,y)\rightarrow(p,q)} f(x,y)= \lim_{x\rightarrow p}(\lim_{y\rightarrow q} f(x,y))$? If so, could someone link me to a proof of it or give me a proof? Edit: So ...
4k views

### What do sine, tan, cos actually mean?

I know that $\sin\theta=\frac{y}{r}$ and $\cos\theta=\frac{x}{r}$. My question is: is $\sin$ a function of $\theta$, as in $\sin (\theta$)? If yes, why is there no $\theta$ on the right hand side of ...
### Is it possible to define $x+x+x+x…x$ times? [duplicate]
Is it possible to define $x+x+x+x...x$ times? I need to compute its derivative. It differs from the derivative of $x^2$. It evaluates to $x$ via sum of derivatives.