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

learn more… | top users | synonyms (1)

1
vote
2answers
64 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
89 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 f^{...
4
votes
3answers
247 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
61 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
303 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
52 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 x^2+b\...
8
votes
2answers
196 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, ...
3
votes
2answers
57 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
89 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
84 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
28 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) $f^{\prime}>0$, $f^{...
1
vote
6answers
193 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 ...
-5
votes
3answers
3k 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
41 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
73 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
98 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 $f,g\in\...
0
votes
1answer
107 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
34 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 $(x_n)$...
2
votes
1answer
76 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
162 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
47 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 ...
1
vote
2answers
75 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 ...
5
votes
1answer
134 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
108 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 ...
7
votes
6answers
613 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
vote
1answer
30 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) & =\text{...
1
vote
1answer
42 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
148 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
0answers
83 views

Prove $X\times Y$ is an equivalence relation [duplicate]

(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 R,...
1
vote
2answers
45 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
55 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 how ...
2
votes
1answer
47 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 x|<\...
2
votes
2answers
66 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 $|x-9|&...
0
votes
1answer
53 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 ...
5
votes
1answer
138 views

Multitangent to a polynomial 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 ...
3
votes
2answers
250 views

Find the inverse function of $\log_{\sqrt{4-x^2} } \left(x^3+5x^2-x \right)$

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
105 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
1k 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 ...
3
votes
0answers
457 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 {x}^{-...
1
vote
1answer
41 views

Looking for a special kind of injective function

Does there exist an injective function $f:\mathbb R \to \mathbb R$ such that for every $c \in \mathbb R$ , there is a real sequence $(x_n)$ such that $\lim\big(f(x_n)\big)=c$ but $f$ is neither ...
3
votes
1answer
122 views

What is the purpose of removable discontinuity?

I've just learned about removable discontinuities. So, when we have such a function we redefine it, making a new function that is defined at the point the first isn't. What is the point of this? What ...
1
vote
4answers
111 views

Show the inverse of a bijective function is bijective

We have a function $\varphi:G\rightarrow H$ is an isomorphism, show its inverse $\varphi^{-1}:H\rightarrow G$ is also an isomorphism I am fine with showing it to be a homomorphism and surjective, ...
1
vote
3answers
73 views

How to answer the question “what is the domain of this function”?

Could you please help me understand and solve this problem about domain of function? All that is written for the question is: What is  the  domain of this function? $$ 2\sin\sqrt{2x-1}+1 $$ ...
3
votes
2answers
3k views

Optimization - find the dimensions of a box as functions of volume - minimal surface area

Had a basic calculus course exam today. This was one of the problems: We have a rectangular box of a given volume V. Present the width, height, and length of the box as functions of V so that the box ...
1
vote
4answers
56 views

Cancelling common factors and equality of functions

Suppose we have two expressions: $\frac{x-1}{x-1}$ and $1$. In the first expression we cancel the nominator and the denominator and are left with $\frac{1}{1} = 1$ and the first two expressions are ...
0
votes
0answers
51 views

What is the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?

Is $1$ the smallest positive period of the fractional part function $\mathbb R \to \mathbb R:x \to\{x \}$ ?
0
votes
1answer
23 views

How to find find $f(x)$ such that $f'(x)=\sin^2(x)$ & $f\left(\frac{\pi}{2}\right)=\frac{\pi}{4}$?

I need to find $f(x)$ such that $f'(x)=\sin^2(x)$ & $f\left(\frac{\pi}{2}\right)=\frac{\pi}{4}$. How to do it?
1
vote
1answer
53 views

How to compute $\int^{1}_{-1}f(x)dx$?

I need to compute $\displaystyle\int^{1}_{-1}\,{\rm f}\left(\, x\,\right)\,{\rm d}x$, where $$ \,{\rm f}\left(\, x\,\right) =\left\{\begin{array}{lcrcl} x & \mbox{if} & x & \leq & 0 \\[...
3
votes
2answers
173 views

Show that $\mathbb{Q}\times \mathbb{Q}$ is denumerable [duplicate]

I am new to functions and relations, and with some concepts I am not so familiar. I have a question in an homework: Show that $\mathbb{Q} \times\mathbb{Q}$ is denumerable. From what I understood,...
2
votes
5answers
432 views

Find value of f(2013)?

Given a function $f(x)$ such that: $f(1) + f(2) + f(3)+\cdots+f(n) = n^2f(n)$ Find the value of $f(2013)$. It is given that $f(1) = 2014$. I tried attempting the question as a bottom-up DP, but ...