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

learn more… | top users | synonyms (1)

1
vote
1answer
17 views

Continuity of a function in a mixed discrete-connected domain

Consider the following function $f: \mathbb{R} \times \left\{0,1\right\} \rightarrow \mathbb{R}$ $$ f(x,y)=\begin{cases}x, & (y=0) \\ k, & (y=1) \end{cases} $$ where $k$ is some real ...
0
votes
2answers
49 views

Why does an $n$th degree polynomial have at most $n-1$ turning points?

How can one explain that polynomial of degree $n$ can have up to $n-1$ turning points and $n$ intersections with the $x$-axis? If it is easier to explain, why can't a cubic function have three or ...
-2
votes
2answers
49 views

Prove that a given equation has solution in specific interval. [on hold]

Prove that the following equation has solution in specific interval. $$2x - 3^{-x} = 0, \quad\mbox{for}\ 0 ≤ x ≤ 1$$ How do I solve this?
0
votes
1answer
22 views

Lebesgue integral, integer part x

$$ \int_{0}^{\infty} 10^{-2[x]} dx $$ How to solve it? is the Lebesgue integral. I drew a graph, it is piecewise continuous. Sum of this function will converge. But I can not understand how it all ...
1
vote
0answers
19 views

Constructing set of functions that give a good basis after a certain integral

I need a set of functions that can be used as a basis after a specific integration. In other words: I integrate a set of functions enumerated by a parameter $k$ where the integral depends on another ...
9
votes
0answers
56 views

$f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$.

Let $f\in\mathcal{C}^2(\Bbb{R},\Bbb{R})$ be a positive function such that $f=\underset{+\infty}{\mathcal{O}}\bigr(f''\bigl)$ does it implies that $f=\underset{+\infty}{\mathcal{O}}\bigr(f'\bigl)$? ...
0
votes
2answers
41 views

To show set is closed and bounded in $\mathbb{R}$

I am having problem with this question , kindly please help me with this , Let $$S = \{x : x^{6} -x^{5} \leq 100\}$$ And $$T =\{x^{2} - 2x : x \in ( 0, \infty)\}$$ Then I have to show that set $S ...
0
votes
2answers
42 views

Finding inverse of a composite function

Let $f (x) = x^{3}+x$ and $g (x) =x^{3} -x$ for all x. I have to find derivative of $g\circ f^{-1}$ at $x=2$. My textbook did this: $(g \circ f^{-1})' (2) = \lim \limits_{h \to 0} \dfrac{g \circ ...
0
votes
0answers
34 views

finding formula from table

I'm implementing a sudoku solver using human way algorithm. Which have 3 constraint, different number ini row, cell and box. I googled and I got ...
3
votes
3answers
67 views

what is meant by $ f ∈ C^{2}[a, b] ?$

What is the meaning of $ f ∈ C^{2}[a, b] ?$ I think it says that $f$ is twice differential on $[a,b]$, isn't it?
0
votes
0answers
38 views

Algorithm-generating algorithm

Is there an algorithm that can create other algorithms based on any number of arguments? For example, a way to determine a function $ f (x) $ from a given input and a given output? I.e. if $ f (2)=4 $ ...
3
votes
1answer
88 views

Is $f(x)=0$ a polynomial function?

Is $f(x)=0$ a polynomial function? we know that constant functions are polynomials of degree zero But, does $f(x)=0$ follow this definition?
0
votes
3answers
42 views

Linear independence related with functions

Good day ! I don't understand the following problem: "Prove that the three functions $x^2,\cos{x},e^x$ are linearly independent" So I think so I have to prove that the linear combination: $a ...
5
votes
2answers
115 views

$f(x+1)-f(x)$ converges $\Rightarrow\frac{f(x)}x$ converges

Let $f:\mathbb{R}\to\mathbb{R}$ continuous such that $\lim_{x\to\infty}f(x+1)-f(x)=l$. How can I prove that $\dfrac{f(x)}x$ converges for $x\to+\infty$ ? I am just trying to prove the convergence, ...
2
votes
2answers
24 views

Function with a set number of pre-images

Let $A,B\subset\mathbb{R},n\ge2$. Let $f:A\to B$ (not necessarily continuous) such that $\forall a\in A,f^{-1}(a)$ is a tuple of $n$ elements. I know that if $f$ in continuous, for $A=B=\mathbb{R}$ ...
1
vote
1answer
38 views

What's the exact definition of symmetry?

My question is basically the same as this, but I haven't found the answer given satisfying. The definition of symmetry I've come up with is this: Let $X \subseteq R^2$. A symmetry of $X$ is an ...
0
votes
1answer
25 views

Prove that a function is total, surjective, injective and find its domain of definition

Let $D = \{1,2,3,4,5,6,7,8,9,10\}$ Let $f:P(N) \to P(N)$, $f(B) = B \triangle D$ I said that the image of this function is: $P(N)$, is that right? It's pretty clear that this function is total ...
1
vote
1answer
16 views

Examples of functions with properties

I'm looking for example of functions defined on $[0,+\infty)$ with the following properties: 1) continuous, twice differentiable 2) $f(0)=0$, $\lim_{x\to+\infty} f(x)=1/3$ 3) $f^{\prime}>0$, ...
1
vote
5answers
90 views

To show that $f (x) = | \cos x | + |\sin x |$ is not one one and onto and not differentiable

Let $f : \mathbb{R} \longrightarrow [0,2]$ be defined by $f (x) = | \cos x | + |\sin x |$. I need to show that $f$ is not one one and onto. I have only intuitive idea that $\cos x$ is even function so ...
-4
votes
3answers
47 views

20 / 5 (2 * 2) can equal multiple answers? [duplicate]

20 / 5 (2 * 2)=1 20 / 5 (2 * 2)=16 It states both answers are equally correct? Is this correct? If so why? If not why? Could this be a matter of perception in the way the person reads the problem and ...
1
vote
1answer
27 views

The alternating sum of primes defines an injection

Define $\displaystyle\alpha(n)=\sum^n_{k=1}(-1)^{n-k}p_k$, where $p_k$ is the $k$:th prime. Show that $\alpha$ is an injection $\mathbb Z_+\to\mathbb Z_+$. It's easy to see while considering sums as ...
3
votes
2answers
45 views

Find $f(x, y)$ when $f(2x + 1, 3y -1) = 4x^2 + 9y^2 + 4x - 6y + 2$

Find $f(x, y)$ when $f(2x + 1, 3y -1) = 4x^2 + 9y^2 + 4x - 6y + 2$ I don't understand, how can we pass two things to a function? Can somebody explain what is this function, please?
1
vote
3answers
86 views

function such that $f(x\cdot t)=f(x)g(t)$

Let $E$ be the set of tuples of continuous functions $f,g:\mathbb{R}^*_+\rightarrow\mathbb{R}$ s.t. $f,g$ are never $0$ and $\forall x,t>0,f(x\cdot t)=f(x)g(t)$. I need to show that ...
0
votes
1answer
89 views

proof for zero function

I am given the following: Let $f$ be a real function, which itself and all its derivatives at $0$ are $0$. Assume there exists $b>0$ such that for all real $x$ and all natural $n$: $|f^{n}(x)|\leq ...
0
votes
1answer
32 views

Looking for another special kind of injective function

Relating to this Looking for a special kind of injective function Does there exist an injective function $f:\mathbb R→\mathbb R$ such that for every $c∈\mathbb R$ , there is a real sequence ...
2
votes
1answer
45 views

Is there an unbounded integrable function with integrable derivative in $(0,1)$?

I wonder if there is a differentiable unbounded function $f\in L^1(0,1)$ with $f'\in L^1(0,1)$. The elementary examples $x^\alpha$ or $\log x$ suggest that my question should be answered negatively. ...
0
votes
3answers
37 views

Sum of functions bounded between 0 and 1?

Let $y \in \mathbb{R}$ and $x \in \mathbb{R}$. I'm looking for two functions $f,g$ such that $$ f(x)+g(y) \in [0,1] $$ Do they exist? In positive case, do you have suggestions for what $f$ and $g$ ...
1
vote
2answers
33 views

To find the number of zeroes

P1 - Given $f (x) = x^ {3} + ax^ {2} +6x -1$ has critical point at $x=-2$, then how many real solutions has $f (x) =0$? MY Attempt regarding is that first i have found value of a which is 9\2 by ...
0
votes
2answers
43 views

X<5,Y<5 (clear)..but what if X<5, Y-X>10

I'm trying to construct geometric representation of the following: X<5, Y<5 (that is clear, it will be the area (square) with the corners on the 5s on X and Y axes. But I am clueless how to ...
4
votes
1answer
81 views

Closed form for sine graphic rotated by 45 degrees?

Is there a non-parametric closed form for a function looking like a sine rotated 45 degrees? I have encountered also a similar question but it asks for a function resembling the rotated sine, but not ...
1
vote
1answer
45 views

One-to-one functions of 2 variables

Are there any one-to-one functions of 2 variables? For each of the following prove or disprove whether there is a one-to-one function $f$ of 2 variables: $f$ is from $\Bbb{N}^2$ to $\Bbb{N}$ $f$ is ...
0
votes
1answer
34 views

representation for Banach algebra [on hold]

How we can represent any Banach algebra as a subspace or Subalgebra of Cb(X)?( in isometrically isomorphic concept) Cb(X)= the set of all complex-valued, bounded , continuous functions on X
7
votes
6answers
268 views

Removable discontinuity question

A quick and easy question on terminology: If we have a function $$f(x) = \frac{x(x-2)}{(x-2)}$$ then clearly the function is not defined at $x = 2$. If we cancel out "$(x-2)$" then the function is ...
-1
votes
1answer
40 views

Suppose all we know about y=f(x) is that it is continuous for all x and f(4)=5. Which must be true?

Suppose all we know about $y=f(x)$ is that it is continuous for all $x$ and $f(4)=5$. Which must be true? a. $f'(4)=5$ b. Every number x is in the domain of f c. The function is increasing near x=4 ...
0
votes
1answer
17 views

Superfunctions with complex iteration indices: Interpretation

Superfunctions are a fascinating concept, allowing us to generalize functional iteration to arbitrary real and complex iteration indices. We have $$ \begin{equation} \begin{split} S_f(0) & ...
0
votes
1answer
33 views

Limit on a continuous differential equation

Let $f$ be a continuously differentiable function and let $L=\lim_{x\to\infty}(f(x)+f'(x))$ be finite. Does this imply that if $$\lim_{x\to\infty} f'(x)$$ exists, then it is equal to $0$?
0
votes
3answers
140 views

Function with zero description

Is there a nice expression (possibly differentiable outside $0$) for a function $f(x)$ that satisfies the following property other than the delta? $$f(x)=1\iff x=0$$ $$f(x)=0\iff x\neq0$$ Is it ...
1
vote
1answer
53 views

Proving this equivalence relation

If $X,Y$ are reflexive, symmetric, and transitive, then $X \times Y$ is an equivalence relation where ${(a,b):a\in X, b\in Y}$. I am trying to self learn these topics. I do know what an ...
0
votes
0answers
57 views

Prove $X\times Y$ is an equivalence relation

(Relation between two sets) If $X$ and $Y$ are sets, a relation between $X$ and $Y$ is a subset $R \subset X \times Y.$ For a relation $R \subset X\times Y$ and $a \in X$ and $b \in Y$ if $(x,y) \in ...
0
votes
2answers
61 views

How do I take the 100th derivative of a polynomial [on hold]

How could I find $$f^{100}(x)$$ for $$f(x)=2x^{100}-7x^{80}+15x^{60}-27x^{40}-18x^{20}+300$$
0
votes
2answers
36 views

Is there any standard terminology for this property?

Let $f$ be a map whose domain is $X$. If $f$ satisfies the property that for all $x\in X$, $$f(f(x))=f(x)\text{,}$$ is there any standard name for such a function? Not sure if "projection" is the ...
0
votes
3answers
44 views

Proving by Cauchy's definition $\lim_{x\to -1} x^2+3x-5=-7$

Prove by Cauchy's definition $\displaystyle\lim_{x\to -1} x^2+3x-5=-7$ From definition: $|x+1|<\delta\Rightarrow |x^2+3x+2|<\epsilon \iff |x+1||x+2|<\epsilon$. Now I'm not really sure ...
2
votes
1answer
28 views

Proving by Cauchy's definition $\lim_{x\to 0} x^2\cos x=0$

Prove by definition that $$\displaystyle\lim_{x\to 0} x^2\cos x=0$$ So take $\delta=\sqrt\epsilon$, and from definition we have: $|x|<\delta\Rightarrow|x^2|<\delta^2\Rightarrow|x^2\cos ...
2
votes
2answers
52 views

Proving $\lim_{x\to9}\sqrt x=3$ using Cauchy's definition

Prove: $\displaystyle\lim_{x\to9}\sqrt x=3$ using Cauchy's definition for a limit. After doing the scratch work I get that: $\delta=\epsilon^2+6\epsilon$, so going back, I have to show that ...
0
votes
1answer
30 views

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A.

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A. I know it looks simple, but my reasoning does not agree ...
0
votes
0answers
14 views

Multitangent to a polinomial function

I'm trying to build some exercises on tangents of functions for beginner students in mathematical analysis. In particular I would like to suggest the study of polynomial functions $ y = p (x) $ of ...
2
votes
1answer
114 views
+100

Find the inverse function 3

Find the inverse function for the following function: $$f(x) = \log_{\sqrt{4-x^2} } \left(x^3+5x^2-x \right)$$ Thanks.
2
votes
1answer
48 views

$f$ is surjective iff it has a right inverse: using the axiom of choice and errors in ProofWiki

Paraphrased from Munkres' Topology: Lemma 9.2. Given a collection $\mathcal{A}$ of nonempty sets, there exists a choice function \begin{equation*} f: \mathcal{A} \to \bigcup\limits_{A \in ...
2
votes
1answer
34 views

Finishing a proof: $f$ is injective if and only if it has a left inverse

I've already done a lot of searching (in particular: https://www.proofwiki.org/wiki/Injection_iff_Left_Inverse) to try to prove this statement: $f: A \to B$ is injective if and only if it has a ...
1
vote
0answers
45 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...