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

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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 ...
1
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1answer
35 views

Function that determines angular velocity?

I see that someone posted the same problem a year ago, but the answer didn't quite give enough info. Here's the question: A movie crew is working on a scene that involves filming a car moving at a ...
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
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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
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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
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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
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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
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1answer
24 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 ...
1
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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 ...
0
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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 ...
4
votes
3answers
46 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 ...
1
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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 ...
0
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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 ...
2
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2answers
27 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 ...
8
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2answers
11k views

Proving a function is onto and one to one

I'm reading up on how to prove if a function (represented by a formula) is one-to-one or onto, and I'm having some trouble understanding. To prove if a function is one-to-one, it says that I have to ...
1
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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?
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0answers
20 views
+50

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 ...
0
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1answer
33 views

functions and recursions

The sequence s(k) where k=1,2,3.... satisfies the recursion s(n)=s(n−2)+s(n−3) for n≥4. If s(n) is rewritten in the form s(n)=s(n−1)+S(n0,n1,…) where S(n0,n1,…) is some linear combination of terms ...
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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 ...
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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 ...
1
vote
2answers
63 views

Showing $(f^{-1}∘g^{-1})=(g∘f)^{-1}$

If the functions $f$ and $g$ are both bijections then the in inverse of the composition function $(f∘g)$ will exist. Show that it will be $(f^{-1}∘g^{-1})=(g∘f)^{-1}$ For the proof assume ...
1
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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 ...
-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 ...
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 ...
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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.
0
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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 ...
-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 ...
1
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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) ...
0
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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)
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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
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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 ...
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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
344 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 ...
1
vote
3answers
187 views

Finding the maximum and minimum values of $f(x)=a^x+a^{1/x}$

Let $f(x)=a^x+a^{1/x}\ (x\gt 0)$ where $a\in\mathbb R$ is a constant. Question 1 : What is the maximum value of $f(x)$ for $0\lt a\lt 1$? Question 2 : What is the minimum value of $f(x)$ for ...
0
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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
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1answer
30 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
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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
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 : $m=\lim\limits_{x\to\infty} \dfrac{f(x)}{x}$ ...
1
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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 ...
0
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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 ...
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 ...
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 ...
1
vote
1answer
470 views

Inverse Trig Functions with Double Angle Formulas

I am studying for a quiz tomorrow and one of the sections I am studying involves rewriting quantities as algebraic expressions of $x$. One of the problems I am having trouble with is: $$\sin ...
0
votes
1answer
38 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 ...
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 ...
1
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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 ...