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

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0
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1answer
20 views

Optimal Value & Uniform Distribution

In a simple setting, $w$ is uniformly distributed on $[0,1]$, R is a function of $wd$. I want to find optimal d in this expression, $aR-(d^2-1)/2$. When I try to find out optimal $d$ than it is $0$. ...
0
votes
0answers
13 views

respective values of two functions are “closer than expected”

let f,g be functions from the same finite set into the reals, let d be the mean distance between f(x) and g(x) for x in S, and let D be the mean distance between f(x) and g(y) for x,y in S; then D-d ...
3
votes
2answers
29 views

For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies?

For what real values does $\phi(x):=1+x+ \dots + x^{2m-1}$ take the value $0$? What can you say about the sign as $x$ varies? I need help adding rigor to my observation to create a formal proof. ...
2
votes
5answers
54 views

Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$

Suppose $f: \mathbb{R} \rightarrow \mathbb{R}$ is twice differentiable. Prove that if $f''(x)+25f(x)=0$ then $f(x)=Acos(5x)+Bsin(5x)$ for some constants $A,B$ Consider $g(x):= ...
1
vote
2answers
53 views

Can f have a finite limit at infinity?

The function $ f:\mathbb R\rightarrow \mathbb R$ is differentiable such that $f(0)=0$ and $(1+x^2)f'(x)\geq{1+(f(x))^2}$ for every $ x\in \mathbb R$ . Can $f$ have a finite limit at infinity?
0
votes
1answer
18 views

Number of Linear boolean-functions [closed]

How many linear boolean functions are there, if we have n variable?
0
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1answer
14 views

$f:(2,4)→(1,3)$ where $f(x)=x-[x/2]$ (where $[.]$ is the greatest integer function/floor function),then what will be $f^{-1}(x)$.

Let $f:(2,4)→(1,3)$ where $f(x)=x-[x/2]$ (where $[.]$ is the greatest integer function/floor function),then what will be $f^{-1}(x)$? I can't understand how to manipulate the floor function.Help ...
0
votes
1answer
25 views

Logarith at base 10 as integration

The logarithmic function with base $e$ is the (set theoretic) inverse of exponential function $e\colon \mathbb{R}\rightarrow (0,\infty)$. This can also be defined using integration as: $\log\colon ...
0
votes
0answers
14 views

How to find local extrema of f (p) give us area of triangle A1B1C1

For a right triangle ABC ( angle C = 90) on the rights height CC1 is chosen point P and consider the triangle A1B1C1 (A1 = AP cross BC, B1 = BP cross AC), if p is distance from point P to AB, to find ...
1
vote
2answers
39 views

Power sets and functions

Let $a\colon\mathcal P(\mathbb N)\to\mathbb N$ be the function defined by $a(X)$ equals $0$ if $X$ has infinitely many elements and $a(X)$ equals the number of elements in $X$ if $X$ has finitely many ...
1
vote
1answer
49 views

Is there always an upper limit for which $\int_0^l f(x)\,dx \; < \; \int_0^l xf(x)\,dx,$ is satisfied?

Given a function $f(x)$ which is strictly positive over all positive values of $x$ such that $f(0) = 0$, it makes sense to me by picturing what happens to $f(x)$ when you multiply it by $x$ that there ...
1
vote
1answer
27 views

Composite functions with domain and codomains

a: $ \mathbb{R} \to \mathbb{R}$ defined by $a(x) = (x/2) + 1$ b: $\mathbb{Z} \to \{0,1\}$ defined by $$b(x) = \begin{cases} 1 \qquad \text{ if } x \geq 1 \\ 0 \qquad \text{ if } x \leq 0 ...
2
votes
2answers
28 views

Rationale behind a proof regarding a continuous function and an open ball

can I have the rationale for the first line of this proof? i.e. How did you know to start answering the question in this manner? I am guessing it is because you want to exploit the definition of ...
0
votes
3answers
60 views

List all functions f: {a, b, c} → {0,1 }.

This is a homework problem I have. Can someone just explains what it means, please? I can think of at least a dozen functions off the top of my head, but I think that's too many to be correct since we ...
1
vote
4answers
48 views

Show $ex \leq e^x$ for all $x \in \mathbb{R}$

So far all I have is this: Let $f$ be a function where $f(x)=ex-e^x\leq 0$ $f'(x)=e-e^x \leq 0$, so $f$ is decreasing. I'm stuck here. Can someone help me with the next steps?
1
vote
0answers
14 views

Verify combination of disjoint subsets $C$ and $D$ is onto

Let $C$ and $D$ be disjoint subsets of set $A$ and $f:C→B$ and $g:D→B$. Define a function $h(x)$ as follows: $$ h(x)=\left\{ \begin{array}{c} f(x) \textrm{ if } x∈C \\ g(x) \textrm{ if } x∈D ...
0
votes
1answer
41 views

Functions and range

$$ a \colon \mathbb{R}\setminus\{0\} \to \mathbb{R} \;\text{ defined by }\; a(x)= 6/x \\ b \colon \mathbb{Z} \to \mathbb{R} \;\text{ defined by }\; b(x) = 3x + 1 $$ a) State the range of ...
4
votes
3answers
47 views

Inverse of an ordered pair?

Let $f: A \to B$ be a bijective function where $A = [0, 2\pi)$ and $B$ is the unit circle. Find the inverse of $f(\theta) = (\cos\theta, \sin\theta)$. I don't understand what it means to take the ...
4
votes
0answers
30 views

Proving $f$ cannot be onto

If $f$ maps finite sets $A$ to $B$ and $n(A) < n(B)$, prove that $f$ cannot be onto. Proof by contradiction: If $f: A→B$ and $n(A) < n(B)$, $f$ is onto. Since, by definition of a ...
-5
votes
1answer
35 views

Oil decay at 13%, how long until it is less than 21% of original?

My teacher gave me this problem, and it is very wordy, I don't really even understand what it is asking. First I took 100 and multiplied it by 0.13 subtracting that number from 100 and completing the ...
-1
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0answers
75 views

What is the name of the function $f(x)=1/(1-x)$? [closed]

I only want to know if the function $1/(1-x)$ has any specific name.
-1
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0answers
21 views

Continous frunctions problem

The problem says: f,g:[0;1]->[0,1] ,2 continous functions.They have the property that f(g(x))=g(f(x))). To solve: Both having the property of DARBOUX on the interval ,demonstrate that the numbers "c" ...
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0answers
32 views

What does R[-a,a] represent?

More precisely: $f \in R[-a,a]$. All I could find was related to the symbol $\mathbb R$, but I have never seen it in this particular constellation, and even if it stood for "$\mathbb R$", I wouldn't ...
0
votes
2answers
37 views

Inner Product Space and Linear Mapping Theorem

I'm having some trouble proving the following theorem: Let $($$X$,$\langle\cdot | \cdot\rangle$$)$ be an inner product space and $f: X \to \mathbb{R}$ a linear mapping. Prove that there exists a ...
0
votes
1answer
32 views

Are these functions equal to each other?

$$f^8(x)=(f^2)^4(x)$$ $$f^8(x)=[f^2(x)]^4$$ Are they equal? I'm confused about these. Please help me clarify this and explain to me why it is so.
0
votes
2answers
25 views

Differential equations in function

Equations (1) : $xy'+(1-x)y=1$ let $z=xy+1$ determine and solve the differential equation (2) whose general solution is the function $z$ . -determine the general solution of (1)
1
vote
1answer
72 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
46 views

Range of function defined as smallest prime divisor. [closed]

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
346 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 ...
1
vote
1answer
34 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
25 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
44 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
44 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
18 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
32 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
29 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
1answer
49 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
28 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
31 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
21 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
39 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
29 views

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

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
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2answers
21 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
vote
1answer
28 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
28 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
vote
2answers
26 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
40 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 ...
1
vote
1answer
27 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 ...