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

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2
votes
2answers
19 views

How to show that $f(x)=2x$ is not onto, as a function from $\mathbb{Z}$ to itself?

$f\colon\Bbb Z\to\Bbb Z$ is given by $f(x) = 2x$. Show that $f$ is one-to-one and not onto. I know this is a very simple question but I can't figure out why this is NOT onto. I've done many other ...
2
votes
0answers
24 views

Identifying a function that involves combinations of terms

I need to know if a function exists that partitions terms in such a way as seen below $$ \frac{d^n}{dx^n}[\frac{(x)_c}{n!}] $$ Note that $(x)_c$ is the falling factorial of x and $c \geq n$, This in ...
5
votes
1answer
183 views

How to find $f(x)$ when $f\left(\frac{x+y}{x-y}\right)\ge\frac{f(x)-f(y)}{f(x)+f(y)}$

We have the function $f(x)$ continuous on $ (-\infty,0)\cup (0,+\infty)$ with $f(1)=-1$ $\displaystyle\lim_{x\to 0}xf(x)=-1$ and $f'(1)$ exsits. For any $x,y\in\mathbb R$ we have ...
1
vote
1answer
27 views

Definition of Point of Inflection

An inflection point is a point on a curve at which the sign of the curvature (i.e. the concavity) changes. According to Wikipedia, "If x is an inflection point for f then the second derivative, ...
0
votes
1answer
55 views

Is $f : A \to P(A), a \mapsto \{a\}$ injective or surjective? [duplicate]

Given an arbitrary set $A$, let $f:A \to P(A)$ be the function defined for all $a \in A$ by "$f(a) = \{a\}$". How would you prove that $f$ is injective or surjective?
0
votes
1answer
16 views

Converting a graph into a function

Is is possible to convert this graph into an function of t(time), if what are the appropriate steps to do this? Any help point in the general direction will be greatly appreciated! Thanks in advance. ...
1
vote
1answer
16 views

Question Regarding Inverses In a Function

Here is my current issue. Our teacher asked a question related to the finding of an inverse of 2. Here is all of the given information: Define "a cross b" as such: a ☢ b = ab + (a + b). Use this ...
0
votes
1answer
25 views

Equation for this linear function

How could I write this function down as an equation? Function is drawn with red color, where k1 and k2 are coefficients which represent the "stepness" of the curve (y=kx+n). This function is purely ...
1
vote
0answers
29 views

Integral equality

I have recently started studying function spaces (namely Lorentz endpoint spaces). Let $\varphi:[0; \infty)\rightarrow\mathbb{R}$ be a non-negative concave function such that ...
-2
votes
1answer
30 views

How to prove that the following function is continuous?

I need to prove that the following function is continuous: $$f(u)=\begin{cases}\dfrac {1-\cos(\sin u)}{1-cos^2u}&,\;\;u \neq 0,\pm \pi,\pm 2\pi,\pm 3\pi,...\\{}\\\dfrac{1}{2}&,\;\;u =0,\pm ...
2
votes
0answers
69 views

Prove that the function is a constant

If a mapping $f:[0,1]\to[0,1]$ is continuous, $f(0)=0$, $f(1)=1$ and $f^n(x)\equiv x$ on $[0,1]$ for some $n\in\mathbb{N}$, then $f(x)\equiv x$. Here ...
-3
votes
1answer
29 views

It is given that $f(x)=x^2-kx$ and $h(x)=f(x+2)-f(x-2)$ [closed]

(a) find $h(x)$ in terms of $k$. (b) If $h(x)=kx-32$, solve $h(x)=40$. How do part B?
0
votes
1answer
19 views

Equation defines y as a function of x?

Hi, I've just moved onto functions in maths and this question really confuses me. Can anyone explain it to me? Please and thanks
-3
votes
1answer
45 views

How to prove that there exist $x \in [1,2]$ such that $2^x=\pi$? [closed]

I need to prove that there exist $x \in [1,2]$ such that $2^x=\pi$ using the continuity of $f(x)=2^x$.
1
vote
2answers
30 views

Can there be two relative minimums on the graph of a quartic function?

I need to find the relative minimum and maximum of the given function: $$x^4-4x^3-2x^2+5x+9$$ Graphing this, I get a relative maximum of $(0.538, 10.6)$ But, I seem to get two relative minimums, at ...
5
votes
2answers
58 views

A Characterization of the Tangent Function?

The tangent function has the amazing property that if $\alpha+\beta+\gamma=\pi$ with $\alpha,\beta,\gamma\in (0,\frac{\pi}{2})\cup(\frac{\pi}{2},\pi)$ then ...
1
vote
1answer
25 views

Oscillation of function and continuity clarification of proof

There is a similar question related to this, but it doesn't answer my question, so I would be thankful if anyone helped me with it. There is a step specifically in the proof I do not understand. ...
2
votes
1answer
20 views

Why $f(t)= t+ 2t^2\ sin (\frac 1t) , t\neq 0$ & $f(t)=0 , t=0$. Prove that this function is not $1-1$ in any neighbourhood of zero.

Why $f(t)= t+ 2t^2 \sin (\frac 1t) , t\neq 0$ & $f(t)=0 , t=0.\,$Prove that this function is not $1-1$ in any neighbourhood of zero. It is not possible with $\frac 1 {n \pi}$.
0
votes
1answer
58 views

sequence get number in sequence from place in sequence

There is a sequence $$X = {1,1,1,1,1,1,1 \dots 2,2,2,2,2,2,2,2 \dots,3,3,3,3,3,3 \dots 4,4,4,4,4,4,4 \dots (k-1),(k-1),k}$$ So there are $(k)$ 1's, $(k-1)$ 2's and $(k-2)$ 3's and so on. Is there ...
1
vote
2answers
41 views

Can I bound the following function by an integer?

I am wondering if the following function $$\frac{x+y}{x^2+y^2}$$ can be bounded in some smart way by an integer.
0
votes
1answer
36 views

Show that if $C(K)$ is separable, then $K$ is metrisable, for $K$ compact and Hausdorff

My question is simply as the title states: Let $(K,\tau)$ be a compact Hausdorff (topological) space. Show that if $C(K)$ is separable, then $K$ is metrisable. Firstly, I appreciate that this is ...
1
vote
4answers
52 views

Write a polynomial equation

Write a polynomial equation with the following characteristics. A quartic function with roots of -3, -1, and 4 (x=4 has a multiplicity of 2) and which passes through the point (5,16) I know how to ...
2
votes
5answers
108 views

What kind of curve is made of half circles?

I have this curve. It's definitely no sine or cosine. It consists of half circles. How do you call it and how do you describe it mathematically?
2
votes
2answers
54 views

Is there a simpler proof that $f(x,y,z) = 4x + 11y + 18z$ is surjective?

So, one could use $f(x,y,0) = 4x + 11y$. Determining if that function is surjective on $\mathbb{Z}$ takes proving that $4x + 11y = c$ has a solution for every $c$. Since we know about the existence ...
5
votes
3answers
252 views

Solving a functional relation

I have this functional relation - $$f\left( x \cdot f(y)\right)=x^2 \cdot y^a$$ which I am trying to solve. I put $x=1$, then I put $f(y)=\dfrac{1}{x}$. I also tried out $y=f^{-1}(1)$, but it ...
0
votes
1answer
29 views

composition of uniformly convergence sequence with continuous function, is uniformly convergence?

Let $(f_n)$ be a series of functions in $C[0,1]$ that uniformly converge to a continuous function $f\in C[0,1]$. a. Let $g: [0,1]\to [0,1]$ be a continuous function. Is it true that $f_n\circ g$ ...
0
votes
1answer
14 views

Linear functions and arithmetic sequences

Suppose we are given two arithmetic progressions: a, a+h, a+2h,... b, b+l, b+2l,... Is it always possible to find a linear function that maps these two ...
3
votes
0answers
41 views

Why can't we generalize straight line equations for all curves?

Apologies, but I'm confused. Let $y = f(x)$ be curve such that at a point $(a,b)$ lying on it, the slope of the tangent kissing that point is $f'(x)$ Now, the equation of the tangent passing through ...
1
vote
1answer
42 views

Question about disproving if $\exists x_0 : f(x_0)=g(x_0)$ then $\exists x_1 : f(x_1)>g(x_1)$

Let $f,g : \mathbb R \to \mathbb R$ such that $f$ is monotone increasing and $g$ is monotone decreasing. Prove/disprove that if $\exists x_0 : f(x_0)=g(x_0)$ then $\exists x_1 : ...
0
votes
1answer
24 views

solving T(n) = 4T(n/2) + n^3 + n*(log(n))^2

I am trying to solve a recurrence by using substitution method. The recurrence relation is: $$T(n) = 4T\left(\frac n2\right) + n^3 + n\cdot\log^2 n$$ i tried to solve this also with master method... ...
1
vote
1answer
27 views

Show that if $\lim_{x\rightarrow\infty}f(x)=a \ \text{and} \ \lim_{x\rightarrow\infty}f'(x)=b$ then $b=0$

Consider: $$f:[c,\infty] \rightarrow\mathbb{R}$$ Show that if:$$\lim_{x\rightarrow\infty}f(x)=a \ \text{and} \ \lim_{x\rightarrow\infty}f'(x)=b$$ Then: $$b=0$$
2
votes
2answers
38 views

Infinitely many times differentiable function with unbounded derivatives?

Let $f$ be an infinitely many times continuously differentiable function on the compact interval $[0,1]$. We denote by $f^{(k)}$ the $k$-th derivative with respect to $x$. Then we know: $\sup_{x \in ...
0
votes
2answers
50 views

Check if a relation on a set is a function [duplicate]

What do I need to look for in order to tell if a relation on a set is a function? Can somebody provide some advice? For example, the relation is defined by $H$ on $A \times \mathcal P(A)$ for $a ∈ ...
0
votes
2answers
55 views

Arbitrary Set A a Function??

Assume you have an arbitrary set A, let RA be the relation defined on A × Power Set(A) by, for all a ∈ A and B ⊆ A, "a RA B iff a ∈ B" 1.Let A = {0, 1}. Is RA a function? Justify? 2.Find a set A ...
3
votes
2answers
62 views

Find $\lim\limits_{x \to \infty} 2x(\sqrt{x-1} - \sqrt{x+5})$

$\lim\limits_{x \to \infty} 2x(\sqrt{x-1} - \sqrt{x+5})$ For what i've found the part in brackets is an indeterminate form. I've tried to multiply the bracket part by $\frac{\sqrt{x - 1} + ...
0
votes
3answers
51 views

Functions Proof

Consider the function f : N → N defined, for every n ∈ N, by "f(n) = (n + 1)! − 1" 1.Prove that, for every n ∈ N, f(n + 1) > f(n) 2.Prove that f is injective. Can somebody shed some light on ...
0
votes
2answers
33 views

Finding a k value which will satisfy 1 positive and 1 negative solution for a given function

Let $f(x) = x^2-10x+9$. The function has $x$ intercepts at $1$ and $9$. For which values of $k$ does $f(x) = 2x + k$ have $1$ positive root and $1$ negative root? My try: I got $f(x) = x^2-12x + 9 = ...
0
votes
0answers
28 views

Integrations of functions with different inputs

I have the following expression \begin{equation} \int f(x,y)g(y) dy = 0, \quad \forall y \in \mathbb{R} \end{equation} and $g(y)\geq 0$. Are there any conditions on the function $f(x,y)$ that we can ...
1
vote
1answer
48 views

Confused on Injection and Surjection Question - Not sure how to justify

Given an arbitrary set $A$, let $f : A \rightarrow\wp(A)$, be the function defined for all $a\in A$ by "$f(a) = \{a\}$" Is $f$ injective? Is $f$ surjective? I am struggling with this question. ...
3
votes
2answers
143 views

Solving a 2 independent variables (2nd degree) recurrence relation

Changes to the recurrences and definition are changed! See here: $f(n, 1) = 2n^2 $ and $f (n, k) = 0$ for $k \geq 2n$ and for $k < 0$ and $f(n, 2n-1) = 1$ for all $n$. Question: Is it possible ...
6
votes
1answer
57 views

Function for which it is unknown whether it is continuous

Is there any function $f:\mathbb R\rightarrow \mathbb R$ for which at least some values are known but it is unknown whether $f$ is continuous or not? Edit: I am looking for examples from actual ...
14
votes
4answers
105 views

High School Advanced Functions: Clarifying log rules in a log equation - $\log(x^2) = 2$, Solve for x.

I got in an argument with my teacher for the possible solutions of x. From some sources i found that because x is squared, negative values should be possible; however, my teacher insists that: $$ ...
1
vote
1answer
28 views

Is there a function $f(x,y)$ such that $f(x,y)$ is unique for all positive integer values $x$ and $y$?

Is there a function $f(x,y)$ such that $f(x,y)$ is unique for $x > 0, y > 0$ and $x$ and $y$ are both integers. Also, how do you prove that $f(x,y)$ is indeed unique?
0
votes
2answers
42 views

Show that there are no $f_1, f_2$ such that $f_1f_2 = f$.

Let $\mathbb{F}$ a finite field. Show that there's $f\in\mathbb{F}[x]$ such that $\deg(f)=2$ and there are no linear polynomials $f_1,f_2\in \mathbb{F}[x]$ such that $f_1f_2 = f$. Hint: Define ...
0
votes
0answers
11 views

Dirichlet Function Integrable bad proof

I am doing a bad proof on purpose for the Dirichlet function, where D(x) =1 if x is irrational, and D(x) is zero if x is rational. I am not working in Riemann integrals, I am working with the poor ...
-1
votes
1answer
44 views

What are the applications of increasing and decreasing functions?

What are some real life applications of increasing and decreasing functions?
4
votes
3answers
208 views

Is inclusion map not the same as identity?

Wikipedia says $i$ is an inclusion means $i: A \to B$ with $A \subset B$ means $i(x) = x$ for each $x\in A$. But doesn't this mean $i(A) = A$, so this is actually identity?
0
votes
0answers
36 views

Formula to divide (group) numbers into N proportionally groups

Lets take ideal theoretical case as example: we have 20 numbers: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20 we should split these numbers into 5 groups: lowest, low, ...
0
votes
0answers
17 views

Finding when $(x^3 + …)/(x^4 + …)$ reaches $10^{-n}$

A problem I’m working on throws up equations like: $$ \frac{1}{4k + 9} + 3\frac{k+1}{(4k + 9)^2} + \frac{2k+1}{(4k + 9)^3} + \frac{k}{(4k + 9)^4} $$ I need to know the value of $k$ for which this ...
0
votes
1answer
25 views

Fraction of a square area within two lines

I have a general line $r: ax + by + c = 0$ and two parallel lines $s,t$ distant $d$ from $r$. And a square of side $l = 1$ centered at $(x_c,y_c)$. The square sides are perpendicular to the $x$ and ...