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

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9answers
401 views

Nonpiecewise Function Defined at a Point but Not Continuous There

I make a big fuss that my calculus students provide a "continuity argument" to evaluate limits such as $\lim_{x \rightarrow 0} 2x + 1$, by which I mean they should tell me that $2x+1$ is a polynomial, ...
7
votes
6answers
213 views

Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$

Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function. Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
7
votes
2answers
241 views

Does $F(t)F'(t) \le 0$ imply that $F$ does not change signs?

The question is in the title, really. More precisely, suppose $F:[0,1] \to \mathbb{R}$ is continuously differentiable and satisfies (i) $F(t)F'(t) \le 0$ for all $t \in [0,1]$ (ii) $F(0) = 1$ ...
7
votes
4answers
654 views

Solving the functional equation $f(x+1) - f(x-1) = g(x)$

Given a function $g(x)$, is it possible to find a function $f(x)$ that satisfies $$ f(x+1) - f(x-1) = g(x) $$
7
votes
3answers
178 views

Is every function $f : \mathbb N \to \mathbb N$ a composition $f = g\circ g$?

True or wrong: For every function $f: \mathbb N \rightarrow \mathbb N$ there is a function $g: \mathbb N \rightarrow \mathbb N$ with $f=g \circ g$.
7
votes
2answers
252 views

Is there a non-constant smooth function mapping $\mathbb{R}$ into $\mathbb{Q}$

I cannot think of a non-constant smooth function which maps all real numbers into rational numbers. Can anyone give a simple example ? The simpler, the better !
7
votes
6answers
399 views

Rigorous Definition of “Function of”

When I was learning statistics I noticed that a lot of things in the textbook I was using were phrased in vague terms of "this is a function of that" e.g. a statistic is a function of a sample from a ...
7
votes
3answers
361 views

What's the difference between a bijection and an isomorphism? [duplicate]

I don't understand what the difference is between a bijection and an isomorphism. They seem to both just be a invertible mapping. Is the set of all bijections a subset of isomorphisms? Or vice ...
7
votes
3answers
356 views

If $f:N→N$ such that $f(f(x))=3x$, then find $f(2013)$

If $f:N→N$ such that $f(f(x))=3x$, then what is the value of $f(2013)$?
7
votes
2answers
43k views

Types of polynomial functions. Quadratic, cubic, quartic, quintic, …,?

I would very much like to have a complete list of the types of polynomial functions. I know that theres: ...
7
votes
3answers
2k views

Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions sin(h),cos(h), and tan(h)?

In matters of conic sections, are there other properties such that it helps to group the circle and hyperbola in one, and the parabola and ellipse in the other?
7
votes
3answers
457 views

Can the exponential function be reprsented as infinite product?

Is there any representation of the exponentil function as infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. $$\mathrm ...
7
votes
1answer
644 views

What is meant by “m|n”? Two letters separated by a vertical bar (|)

I am new to this subject, and not not sure what "|" symbol means on this statement. Let $R_2 \subset\Bbb N \times\Bbb N$ be defined by $(m, n) \in R_2$ if and only if $m|n$.
7
votes
4answers
284 views

Mustn't a function map every element of its domain to range (but not codomain)? [Richard Hammack, P228]

How to Prove It, D Velleman P226, P228: Suppose $f$ is a relation from $A$ to $B.$ Then $f$ is a function from $A$ to $B$ means: $\forall \; \color{#009900}{a \in A}, \exists \; ! \; b \in ...
7
votes
3answers
357 views

Finite at every point but unbounded on every interval

Is is possible that a function $f$ is finite at every point but unbounded on every interval? What if f is measurable?
7
votes
1answer
146 views

$f(x)^2 ≥ f(x + y)(f(x) + y)$ for no $f$?

Prove that there is no function $f : \mathbb{R}^+ → \mathbb{R}^+$ such that $$f(x)^2 ≥ f(x + y)(f(x) + y)$$ for all $x, y > 0$. I can't think of a way of solving this.
7
votes
1answer
323 views

Can functions have multiple inputs?

Now bear with me here, I'm not the best at math. I'm just trying to find out something that I never really learned. I was wondering, can a function have multiple inputs such as this one below? ...
7
votes
2answers
238 views

Prove that between two roots of $f(x)$ there is a root of $g(x)$

Let $f(x),g(x)$ be differential functions, and $f'(x)g(x)\neq f(x)g'(x)$ for all $x\in\mathbb R$. Prove that between two roots of $f(x)$ there is a root of $g(x)$. I guess this has to do with Rolle's ...
7
votes
3answers
252 views

Category theory without codomains?

A surjection is a function whose range equals its codomain. Thus, the distinction between functions and surjections requires the notion of a codomain. Similarly, a bijection is an injection whose ...
7
votes
4answers
664 views

Proof of linear independence of $e^{at}$

Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$, form a linearly independent set ...
7
votes
1answer
209 views

finding the value of $f(2001) $ if…

if $f (\frac{x}{y}) =\frac{f(x)}{y} $ and $f(2000)=1$ ; then what's the value of $f(2001)$. I tried hard but can't figured out anything. please help me, how can I proceed?
7
votes
4answers
296 views

Why does $\ln(x) = \epsilon x$ have 2 solutions?

I was working on a problem involving perturbation methods and it asked me to sketch the graph of $\ln(x) = \epsilon x$ and explain why it must have 2 solutions. Clearly there is a solution near $x=1$ ...
7
votes
1answer
550 views

Higher Order Trigonometric Function

Once in a time, I had to work with functions that have the following Taylor series expansion: $$ t_m(x)=1-\frac{x^m}{m!}+\frac{x^{2m}}{(2m)!}+\cdots =\sum_{k=0}^\infty \frac{(-1)^k x^{km}}{(km)!}. $$ ...
7
votes
2answers
102 views

Function such that $f(x) f(\pi/2 - x) = 1$

I'm looking for functions that are smooth ($C^\infty$) between $0 < x < \pi/2$ that satisfy the equation $$f(x)\, f(\pi/2-x) = 1$$ on the inteverval $0<x<\pi/2$. I know that the constant ...
7
votes
2answers
75 views

Prove that there exists a sequence $\{x_{n}\}$ such that for every $n\,\quad f_{n}$ has a global maximum

For every positive integer $n$ consider function $f_{n}(x)=n^{\sin x}+n^{\cos x},\ x \in \mathbb{R}$. Prove that there exists a sequence $\{x_{n}\}$ such that for every $n,\ f_{n}$ has a ...
7
votes
3answers
257 views

Show that $\frac{(1-y^{-1})^2}{(1-y^{-1 -c})(1-y^{-1+c} )}$ is decreasing in $y > 1 $.

I am interested in the function $f(y) = \frac{(1-y^{-1})^2}{(1-y^{-1 -c})(1-y^{-1+c} )},$ for values of $c \in (0,1)$, and $y > 1$, and have been trying to show that the function is decreasing. I ...
7
votes
1answer
561 views

Can a function be increasing *at a point*?

From what I understand we say that a function is increasing on an interval $I$ if $$ x_1 < x_2 \quad\Rightarrow\quad f(x_1) < f(x_2). $$ for all $x_1,x_2\in I$. I understand that some might call ...
7
votes
5answers
146 views

Composition of Inverse Functions

$f$ and $g$ are inverses of each other when $f(g(x)) = x = g(f(x))$. However, can there be 2 functions where $f(g(x)) = x$ but $g(f(x))$ does not equal to $x$? I feel like there are but I cannot find ...
7
votes
2answers
129 views

Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$ f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0. $$ Can someone give explicit ...
7
votes
2answers
105 views

How to find period of $f$ if $f(x+13) + f(x+630) = 0$

Let $f:\Bbb{R}\to\Bbb{R}$ be a periodic function with period $T$. The question was to find the (fundamental) period given the following relation. $$ f(x+13) + f(x+630) = 0 $$ Now, the given method ...
7
votes
1answer
701 views

Show f is uniformly continuous on $(a,b)$ if it is continuous and $\lim\limits_{x\to a^+}f(x)$ and $\lim\limits_{x\to b^-}f(x)$ exist

Let $f:(a,b)\to\mathbb{R}$ be continuous at all $x\in(a,b)$. If $\lim\limits_{x\to b^-}f(x)$ and $\lim\limits_{x\to a^+}f(x)$ exist in $\mathbb R$, how can we prove that $f$ is uniformly continuous on ...
7
votes
1answer
333 views

Proving or disproving $f(n)-f(n-1)\le n, \forall n \gt 1$, for a recursive function with floors.

The Olympiad-style question I was given was as follows: A function $f:\mathbb{N}\to\mathbb{N}$ is defined by $f(1)=1$ and for $n>1$, by: ...
7
votes
2answers
244 views

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities?

Is the use of $\lfloor x\rfloor$ legitimate to correct discontinuities? In functions like $\tan^{-1}(a \tan(x))$, the angle wraps and the result is discontinuous. Is it legitimate to redefine the ...
7
votes
1answer
235 views

When to create transcendental function to solve “unsolvable problem”?

$\int \frac{1}{x} dx$ is an unsolvable problem using standard laws of Calculus (power rule) without the use of the function $f(x) = \ln x$ which was handcrafted by mathematicians to solve such ...
7
votes
1answer
429 views

Find $f(5)$ of a non-constant polynomial function $f(x)$

Suppose $f(x)$ is a non-constant polynomial such that $f(x^ 3) − f(x ^ 3 − 2) = f( x )\cdot f(x) + 12$ for all $x$. Find $f(5)$? I find this problem on Quora just now, and I try to solve it but do ...
7
votes
1answer
268 views

Intuition behind convex functions

For me, possibly the most out-of-nowhere definition of the first semester of Calculus was the following definition of a convex function and its equivalents. Function $f$ is convex on the interval ...
7
votes
2answers
162 views

Prove that no function exists such that…

The exercise goes like this: Find a continous function $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall c \in \mathbb{R}$ the equation $f(x)=c$ has exactly 3 solutions; Prove that no ...
7
votes
3answers
121 views

How can we prove this function must be linear?

Let $f:[a,b]\to\mathbb{R}$ be continuous. Suppose for any sequence $(r_n)_{n=0}^{\infty}$ with $\lim_{n\to\infty}r_n=0$, and any $x\in(a,b)$: ...
7
votes
2answers
107 views

an injection into $\mathbb{N}$

Is that true that the map $f\colon \{(m,n)\in\mathbb N^2:m\le n\}\to\mathbb N$ defined by $(m,n)\mapsto (m+n)^{\max\{m,n\}}$ is an injection? If it is, how to prove that? I have asked a similar ...
7
votes
1answer
235 views

$C$ be a closed subset of the Cantor set $\Delta$. Show the existence of a continuous function $f:\Delta\to C$ s.t. $f(x)=x$, $x\in C$

Question: Let $C$ be a closed subset of the Cantor set $\Delta$. Prove there is a continuous function $f$ from $\Delta$ onto $C$ s.t. for every $x \in C$ we have $f(x)=x$. Context: Advanced ...
7
votes
2answers
547 views

Proving Injectivity

The problem is to show the function $f:\mathbb{R}^2\rightarrow\mathbb{R}^2$ given by $$f(x,y)=(\tfrac{1}{2}x^2+y^2+2y,\,x^2-2x+y^3)$$ is injective on the set ...
7
votes
2answers
61 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
7
votes
1answer
169 views

Integer functions

For $x>0$ consider the following three functions: $$\begin{align} f(x)&=x+1;\\g(x)&=2x;\\t(x)&=3x \end{align}$$ Let $A(x)$ be the minimum number of operations using only functions ...
7
votes
1answer
175 views

Determine an explicit expression for $f$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, bounded such that the space $\mathrm{lin}\{f_k(x)=f(x+k)∣k ∈\mathbb{N}\}$ is finite-dimensional. Determine an explicit ...
7
votes
1answer
1k views

Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.

Given any set $X$, let $P(X)$ be the set of all subsets of $X$, and let $\{0,1\}^X$ be the set of all functions $X \rightarrow \{0,1\}$. Construct a bijection (and its inverse) between P(X) and ...
7
votes
1answer
204 views

A question about showing $f(x)=0$

Let $f$ be a function from the set of real numbers to itself that satisfies $f(x + y) ≤ yf(x) + f(f(x))$ for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x ≤ 0$. I tried to show that ...
7
votes
0answers
328 views

Compositional “square roots”

Let $A$ be a set and $f\colon A\to A$ a function. The primary and general question is: What conditions are necessary so that there exists $g$ such that for each $x$ in $A$, $g(g(x))=f(x)$? This is a ...
7
votes
0answers
294 views

Does this calculation have a name, or a generic formulation?

Background I would appreciate help in identifying / explaining this operation: To calculate each of the $n$ values of $f(\Phi)$: sample from the distribution of each of $i$ parameters, $\phi_i$ ...
7
votes
0answers
205 views

When does the “Zetor function” converge?

Let $p_n$ be the n'th non-trivial zero of the Riemann zeta function. We define the Zetor function (acronym of 'zeta' and 'zero') as follows: $$\zeta \rho (s) = \sum_{n=1}^{\infty} \frac{1}{(p_n)^s}. ...
6
votes
10answers
1k views

how to see the logarithm as the inverse function of the exponential?

I saw here in math.stackexchange some proofs of how the log and exp functions are related to each other, but I want to get an intuition for that. In layman terms, how would you explain the connection ...