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

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1
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1answer
53 views

Can a simple but rigorous argument be found to prove that this function is strictly increasing?

I have a problem here that asks to show that the function $ f: [0,\infty) \to \mathbb{R} $ defined by $$ f(x) \stackrel{\text{df}}{=} \begin{cases} \dfrac{1}{x} \left( 1 + \dfrac{x^{2}}{4} \right) ...
-4
votes
2answers
27 views

Range of function defined as smallest prime divisor. [on hold]

Let $P(x) : \{8,9,10,11,12,13,14,15,16\} → N$ be the function defined by $P(x)$ equals the smallest prime number that divides $x$. (a) Write down the set of ordered pairs which corresponds to $P$. ...
3
votes
2answers
170 views

Determine if the following is surjective

I need to determine if $f: \Bbb N\times\Bbb N \to \Bbb N$ such that $f(a,b) = a^b$ is a surjective (onto) function. My intuition is that it is but I don't know how to prove it. I don't even know how ...
0
votes
1answer
30 views

Show that f(x) is convex

Show that f(x) = inf{g(x1)+h(x2)} is convex subject to x1+x2=x and where g(.) and h(.) are convex functions. Can I just go about this by using the regular definition of a convex function or which ...
0
votes
1answer
23 views

Value of a function at Jump Discontinuitiy?

How do you define value of unit step function $t=0$? We know, $ u(t) = 0, t <0\\ $ and $ 1, t>0$ but what should be the value of $u(0)$? I find both $u(0) = 0.5$ and $u(0) = 1$ are used in ...
1
vote
3answers
35 views

Parameterizing cliffs

I am looking for a function $f(x; \alpha, X_1, X_2, Y_1, Y_2)$ that has the following property: For $\alpha=0$ it behaves linearly between $(X_1, Y_1)$ and $(X_2, Y_2)$, and as $\alpha$ gets closer to ...
0
votes
2answers
39 views

Will $y=\sqrt x$ be an into function or onto function?

Will R $f(x)=\sqrt x$ be an into function or onto function? How to understand from the graph that it will be onto or into function (just by looking at the graph) ? Domain:positive real numbers ...
0
votes
1answer
17 views

Show that if $g((x_n)) \rightarrow l$ and $g((y_n)) \rightarrow m$, then $l=m$

Suppose that $g: (a,b] \rightarrow \mathbb{R}$ is uniformly continuous. Suppose that both $(x_n), (y_n)$ are sequence in $(a,b]$ which converge to $a$. Show that if $g(x_n)) \rightarrow l$ and ...
0
votes
1answer
26 views

Show that $f$ is uniformly continuous.

Suppose that $F:(a,b] \rightarrow \mathbb{R}$ is continuous and that the limit as $x \rightarrow a$ of $f(x)$ exists. Show that $f$ is uniformly continuous. I am really struggling with this one. HELP ...
0
votes
2answers
27 views

Finding the asymptote of $\tan(x)$

Using limits to find the asymptote of a function $y=f(x)$ is usually done with limits as : if the asymptote is of the form $y=mx+c$ then : $m=\lim\limits_{x\to\infty} \dfrac{f(x)}{x}$ ...
0
votes
0answers
37 views

Find the function satisfying the given condition

If $f(xy)=e^{xy-x-y}[e^y f(x) +e^x f(y)]$ and $f'(1)=e$. $f'$ denotes the derivative of function $f(x)$. Find $f(x)$. I could find that $f(0)=0$ and $f(1)=0$ and then found the derivative the got ...
0
votes
0answers
22 views

Limit property of a function: $\lim_{p \to 0} \frac{w(c p)}{w(p)} \in (0,\infty)$

I have a function that has (needs to have) the following property: $\lim_{p \to 0} \frac{w(c p)}{w(p)} = k \in (0,\infty)$ for all $c \in (0,\infty)$. Do you know how this property is called or ...
0
votes
1answer
23 views

Best aproximation to an numerical solution using two aproximated functions

I want to find the best aproximation to a numerical solution. For that I want to use two aproximated functions (that I already know). If I plot them I see that one of them underestimates the original ...
2
votes
1answer
18 views

Solving equation with functions inside the function?

I've been given the problem: For h(x) defined below, find h′(2), given that: f(2)=−3, g(2)=3 , f′(2)=−1 and g′(2)=7. h(x) = f(x)g(x) I was thinking h'(x) = (-1)(7) = -7 Is this right? If ...
0
votes
1answer
32 views

Prove carefully that $g(f(x)) \rightarrow l$ as $x \rightarrow x_0$

Suppose that (i) $f(x) \rightarrow y_0$ as $x \rightarrow x_0$, (ii) $g(y) \rightarrow l$ as $y\rightarrow y_0$ and (iii) $g(y_0)=l$. Prove carefully that $g(f(x)) \rightarrow l$ as $x \rightarrow ...
-1
votes
2answers
28 views

Biyection between $Q=\{S\subseteq \mathbb{N}|0\in S\}$ and $\mathbb{P}(\mathbb{N})=\{A\subseteq\mathbb{N}\}$ [on hold]

Can anybody help me find a biyection between $Q=\{S\subseteq \mathbb{N}|0\in S\}$ and $\mathbb{P}\left(\mathbb{N}\right) = \{A\subseteq\mathbb{N}\}$
1
vote
2answers
17 views

Proof DES is injective - is this a valid argument

Without going too much into detail into the crpytography of the matter since not every mathematician is interested or knowledgable in the field, there is an encryption process called DES (data ...
1
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1answer
27 views

Find the fixed points of a function $f(x) := exp(x - 2)$ using a recursive algorithm

I need to find the fixed points (i.e. when $f(x) = x$) of the following function $f(x) := exp(x - 2)$. I understood that the fixed points should be the intersecation points between $f(x)$ and a ...
1
vote
1answer
26 views

Formula for the Beta function for natural m, n

Using only the definition $$B(x, y) = \int_0^1 t^{x-1}(1-t)^{y-1}dt$$ for the Beta function $B(x, y)$, it's symmetry $B(x,y) = B(y,x)$ aswell as the fact that $(x + y)B(x + 1, y) = xB(x, y) ...
1
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2answers
22 views

Monotonous everywhere function

$f: \mathbb R \to \mathbb R,\forall x \in \mathbb R $ $\exists \delta \gt 0 : f$ is non-decreasing on $(x-\delta,x+\delta)$(I call that statement A). I need to prove that $f$ is non-decreasing on ...
1
vote
2answers
39 views

Inverse function.

A function $h$ is defined by $h:x\rightarrow 2-\frac{a}{x}$, where $x\neq 0$ and $a$ is a constant. Given $\frac{1}{2}h^2(2)+h^{-1}(-1)=-1$, find the possible values of $a$. Can someone give me some ...
0
votes
1answer
25 views

Quick question about mod equation

So here is the mod function: $$5 ^ {31} \cdot 2 ^{789} - 23^{23}\pmod{10}$$ Is there a way to shorten it, or I must calculate it plain numbers? I have tried the mod powers rule, but except for the ...
1
vote
2answers
33 views

Showing that a function is strictly increasing

Let $f(x)=x/(1+|x|)$, $x\in\mathbb{R}$ This is a simple question but I am a bit stuck to show directly that $f$ is strictly increasing, so without any tools like the 1st derivative test, so just using ...
0
votes
1answer
22 views

Hypothesis needed for existence of an interval without a function zero

While studying ODE I thought of the following problem: Let $f:A\subset\mathbb{R}\to\mathbb{R}$ and $x_0\in A$ such that $f(x_0)=0$. What properties should have $f$ so as to allow us to conclude that ...
0
votes
1answer
15 views

Functions - Trig - Determine [on hold]

The vertical displacement of the end of a robot arm (in cm) at time t (in [16 marks] seconds) is given by y = 8 + 7 cos 3t + 7 cos 6t: (a) Find all times, t > 0, (in exact form i.e. in terms of ...
0
votes
1answer
20 views

Formula with differing output based on sign of input

I'm looking for a mathematical formula that will reproduce this pseudo-code: if x >= 0 x+=1 else x-=1 If this is possible, what would such a formula look ...
0
votes
1answer
14 views

Logical comparison of two values with algebra

Suppose I have two real numbers A and B (A $\wedge$ B $\subset$ $\mathbb{R}$). I want to do some algebra over these number and get 1 if they are equal and get 0 if not. For example: In this ...
4
votes
1answer
62 views

Need Help Understanding Notation With Functions

Original picture: LaTeX approximation: $$f\color{blue}{\substack{(x)\\x\to\infty}}=\pm\sqrt{\frac{(x^2+x)^3}{\pi}}.$$ What does the notation highlighted in blue mean? I understand that ...
1
vote
3answers
66 views

Finding lower/upper bounds for $\prod_{i=2}^n \log(i)$

I have a homework problem where I need to asymptotically order a set of functions, and $\prod_{i=2}^n \log(i)$ is one of them. Is there a tight upper/lower bound for this function? I've tried the ...
0
votes
1answer
11 views

Prove that the relation $R = \{(f,g)\in F \times F \mid \exists h \in P:(f = h^{-1}\circ g\circ h)\}$ is reflexive.

Prove that the relation $R = \{(f,g)\in F \times F \mid \exists h \in P:(f = h^{-1}\circ g\circ h)\}$ is reflexive, where $F = \{ f \mid f : A \to A\}$ and $P = \{f\in F \mid f\text{ is one-to-one ...
0
votes
1answer
55 views

Primitive of the function $(\sin x)/x$

I know that for some functions, for instance $f(x) = e^{-x^2}$, there does not exist a primitive. Does there is a primitive for the function $f(x) = \frac{\operatorname{sin}(x)}{x}$?
0
votes
1answer
16 views

Matlab noob looking for probably very basic advice to plot a curve [on hold]

Basically just started to use Matlab and am struggling with this basic example. My function or variable d is undefined but am unsure on how to progress. If anyone can guide me through the first ...
0
votes
0answers
28 views

strong convex implies exp-concave

Prove that if f is strong convex (for some m>0) $\mbox(\nabla f(\mathbf{x})-\nabla f(\mathbf{y}))^{T}(\mathbf{x}-\mathbf{y})\geq m||\mathbf{x}-\mathbf{y}||_{2}^{2} $ then f is also ...
0
votes
2answers
41 views

What's the relation between a fixed point and a root of a function?

A fixed point of a function $f$ should be an $x$ in the domain of $f$, such that $f(x) = x$. On the other hand, a root (or zero) of a function, should be an $x$ in the domain of $f$, where $f(x) = ...
-1
votes
0answers
35 views

Find inverse of a double function

I have the following function: $$f(x)=\begin{cases}3x+1,~x\gt 0\\2-x^2,~x\leq 0\end{cases}$$ and I need to find its right inverse. So far I got that, ...
0
votes
0answers
31 views

Terminology for functions such that $f(x)\ge x$ for all $x$

Is there a common terminology for a real function $f$ such that $$f(x)\ge x$$ for all $x$? same question for the conditions $\forall x,f(x)>x$; $\forall x,f(x)\le x$; $\forall x,f(x)<x$. (I'm ...
0
votes
1answer
18 views

Need help in understanding proof of continuity of monotone function

I am reading the following proof of a proposition from Royden+Fitzpatrick, 4th edition, and need help in understanding the last half of the proof. (My comments in italics.) Proposition: Let $A$ be ...
-2
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0answers
27 views

Implicit function theorem and derivatives

I'm a bit confused about determing $y'$ and $z'$ If I differentiate both equations wrt $x$ I get: $2x+2y\frac{dy}{dx}+2z\frac{dz}{dx}=0$ and $y+(x+z)\frac{dy}{dx}+y\frac{dz}{dx}=0$ Now because ...
3
votes
2answers
26 views

What does it mean to write $(y,z)=G(x)$

I understand this is a map from $\mathbb{R} \mapsto \mathbb{R^2}$. Is it the case that $y=y(x)$ and $z=z(x)$? i.e Are $y$ and $z$ individually functions of $x$ as well as being so jointly ...
0
votes
0answers
8 views

References for a notion related to radially lower semicontinuity

Let $E$ be a real vector space, $C\subset E$ be a nonempty convex set and $z\in C$. Let $f:C\rightarrow\mathbb{R}$ such that $$ \textbf{(A)} \quad f(z)\leq\limsup_{t\downarrow 0}f(z+t(w-z))\quad ...
-1
votes
1answer
60 views

Can a function map $\mathbb{R}\mapsto\mathbb{R^2}$ [on hold]

Is the mapping in the title of this question possible?
2
votes
1answer
56 views

Why does the graph of $y=\gcd \left(\frac{x}{y},xy\right)$ seem to have 4 “straight” lines?

Why does the graph of $y=\gcd \left(\frac{x}{y},xy\right)$ seem to have 4 "straight" lines? Using https://www.desmos.com/calculator for plotting.
0
votes
2answers
53 views

Prove or disprove: If $\lim_{n\to\infty} (a_{2n} - a_n) =0$, then $a_n$ has a limit (not infinity). [duplicate]

I need to prove if this is true or false.If true then I need to Prove and if false I need to provide an example that disproves the statement.I tried many times but it didnt work. Note:- Im new so I ...
0
votes
0answers
13 views

How to design the analyticity of a function to find when the variable becomes zero

Suppose there is a n-arity function $f(x_0,x_1,...x_n)$ and I want to use it as fitness function. By optimizing function $f$, I want to know when variable $x_0$ or $x_1,...,$ or $x_n$ comes to zero. ...
1
vote
0answers
13 views

Properties Of Different Kinds Of Functions - Venn Diagram Method [on hold]

Suppose there are two sets A and B.A has m elements and B has n elements. What will be maximum number of 1)Functions 2)One-One Functions 3)Many-One Functions 4)Into Functions 5)Onto Functions ...
1
vote
1answer
23 views

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$]

At how many points will $\lfloor(sin x + cos x )\rfloor$ be discontinuous in the interval [0,2$\pi$] ? How should the graph be ?
0
votes
3answers
64 views

How do I prove this Limit? [on hold]

How do I prove $$\lim_{n\to\infty}(\sqrt[3]{a_n+1}-\sqrt[3]{a_n})=0 $$ where $a_n\to \infty$ using the standard definition of the limit, or in other words using $\epsilon$?
2
votes
1answer
25 views

Smooth function conditions

A curve defined by $x=f(t)$, $y=g(t)$ is smooth if $f′(x)$ and $g′(x)$ are continuous and not simultaneously zero. Why do we have the second condition(simultaneously zero)?
1
vote
0answers
23 views

When is it possible to bound a function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ with $\big|\ f(x_1,x_2,\ldots,x_n)\ \big| \le {\prod}_{i=1}^n h_i(x_i)$

Is there any result that specifies when a multivariate function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ can be bounded (either locally or globally) by a product of some functions $h_i:\mathbb{R} ...
0
votes
0answers
8 views

What is the local form of the function at the point of self-intersection with a contour?

I am trying to solve this question : One of the contours (i.e. loci of locations with the same value) of a generic smooth scalar function of the two-dimensional plane is roughly ...