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

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0answers
32 views

Fourier transform of this oddly worded function

A function is equal to zero outside a unit area square centered at (0,0) and inside a central quarter-unit area square similarly oriented. Elsewhere the function is equal to unity. I am trying to find ...
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0answers
22 views

Linear function y=a(x) equation function

Can someone help me with nr 4 5 6 i got 3rd one = 6
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1answer
37 views

Limits of functions on first countable spaces

Would I be right in stating that for first-countable spaces when finding the limit of a function it can equivalently be reduced to finding a limit involving sequences: For example if looking for the ...
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1answer
36 views

Proving a point is a local minimum

I have a rather basic question. I have a function $f:R \rightarrow R$, and I want to show a point, $x^*$, is local minimum, i.e., $f(x^*+\delta) \geq f(x^*), \ \delta \to 0$. I can show that: $f(x^* ...
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2answers
79 views

Prove the limit $\lim_{x\rightarrow-1^{+}} = \frac{1}{x^{^{2}} -1}$ exists.

For each of the following, use definitions (rather than limit theorems) to prove that the limit exists. Identify the limit in each case. (c) $\lim_{x\rightarrow-1^{+}} = \frac{1}{x^{^{2}} -1}$ ...
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1answer
33 views

Find the limit of √x^2/x as x approaches zero from the left.

For each of the following, use definitions (rather than limit theorems) to prove that the limit exists. Identify the limit in each case. a) lim_x→ 0^(-) = √x^2/x proof: Suppose f(x) = √x^2/x, and L ...
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1answer
20 views

If there exist numbers M and m such that m < f(a) < M, prove that there exist positive numbers ε and δ such that m + ε < f(x) < M - ε

Suppose that a is a real number and I is an open interval which contains a. If f: I → R satisfies f(x) → f(a), as x → a and if there exist numbers M and m such that m < f(a) < M, prove that ...
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4answers
31 views

Compose a function based on a word problem

I'm not looking for answers, I'm just having a hard time with composing a function out of a Max/Min problem like this. Possibly just show me how you would compose the function and leave the rest for ...
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4answers
65 views

Why is $f(x)^{-1}$ used to denote the inverse of a function, and not its reciprocal?

Function notation says that any operations applied to a variable inside the parenthesis are applied to the variable before it enters the function, and anything applied to the function as a whole is ...
2
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2answers
35 views

If L = lim_x→a f(x) exists, then |f(x)| → |L| as x → a .

Suppose that f is a real function. a) Prove that if $L = \lim_{x\to a} f(x)$ exists, then $|f(x)|\to |L|$ as $x \to a$ . Proof: Suppose that f is a real function. And suppose $L = lim_{x\to a} ...
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1answer
20 views

What would the Big oh be of (1/2)^n

Is it just (1/2)^n? The function itself gets closer and closer to 0 as x > infinity but I don't know what its classification would be in terms of big oh. O(1)?
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0answers
28 views

how write these in interval notation

how write these in interval notation 1- all real numbers except $-3/4$ I assume it will be: $(-\infty , -3/4)\cup(-3/4 , \infty)$ 2- all real numbers except $1/4$ I assume it will be: ...
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1answer
19 views

Converting $f_{2}$ to $O(f_{2})$ that isn't $f_{1}$

So the actual problem I'm trying to figure out is Find functions $f_{1}$ and $f_{2}$ such that both $f_{1}(n)$ and $f_{2}(n)$ are $O(g(n))$, but $f_{1}(n)$ is not $O(f_{2})$ I know that if I had ...
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1answer
27 views

Numerically find all zeros of multivariate function

How do I find all zeros of a multivariate function , i.e. f(x1,x2,x2,...xn)=0 numerically? I don't know exact analytic form of f , but can numerically compute f at every point on its domain. ...
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2answers
66 views

Continuous Function of functions

Let $f,g :\mathbb{R} \to \mathbb{R}$ be continous functions. Define $h :\mathbb{R} \to \mathbb{R}\ \ \text{by}\ \ h(x) = 5f(x) -2g(x)$. Show directly from the definition that $h$ is continuous. All ...
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2answers
33 views

How to come up with One-To-One and Onto Examples

I'm trying to come up with example functions that are $N \rightarrow N$ for each category: One-to-one but not onto. Onto but not one-to-one. Nether one-to-one nor onto. Both one-to-one and onto. ...
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1answer
26 views

Function With a Finite Sequence As the Domain?

Here is a prime example of what I have in mind: The prime counting function, pi of x, is equal to 0 when x is 1, it is equal to 1 when x is 2, and it is equal to 2 for both x=3 and x=4. So can I write ...
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1answer
22 views

Sum of cosines in a transmitter

Is there a closed form solution to this sum? $$ \max \sum_{k=1, 2, ...}^n \cos (k+m(k) \pi /4), m(k)= 0, 1, 2, or 3$$ The application where this arises is calculating the peak voltage of a radio ...
4
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1answer
31 views

Why should transcendental functions and their arguments be dimensionless?

While looking for the answer on the internet I came across an answer giving this explanation "Another way of seeing clearly why an exponential's argument should be dimensionless is to Taylor expand: ...
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3answers
41 views

Derivative of a function with respect to another function.

I want to calculate the derivative of a function with respect to, not a variable, but respect to another function. For example: $$g(x)=2f(x)+x+\log[f(x)]$$ I want to compute $$\frac{\mathrm ...
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1answer
13 views

Logarithm Via Multiplication By Some Function

This question may be totally rubbish, I'm not sure... Basically, I'm wondering if I have some expression, say, $$f(z)A(z)$$ where $A(z)$ is some arbitraryly changing function in $z$ totally out of my ...
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1answer
35 views

Proving uniformly convergence on a Banach Space

Let $\phi:\mathbb R\mapsto\mathbb R$ is defined by $\phi(x)=\frac1{\sqrt{2\pi}}e^{-\frac{x^2}2}$ for all $x\in\mathbb R$ and $$ {\cal L}_0^2(\mathbb R)=\left\{f:\mathbb R\mapsto\mathbb R\ |\ ...
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6answers
209 views

Find the range of $f(x) =(x-1)^{1/2} + 2\cdot(3-x)^{1/2}$

How to take out the range of the following function : $$f(x) =(x-1)^{1/2} + 2\cdot(3-x)^{1/2}$$ I am new to functions hence couldn't come up with a solution.
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0answers
36 views

Find the solution for a equation.

Please help me to find the solution x of the following equation, with $a$, $T$ are constants and $0<x<T/4$: $(1-4x/T)(1+a/x)\ln(1+a/x)-a/x+2a/T=0$ P/s: Actually, I have tried to convert this ...
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1answer
21 views

Determine whether the graph of the function is the graph of a one-to-one function.

the graphs in #44, 48, 63, 65, 68 For each graph, does it represent a one-to-one function? my solution are: 44_ Yes _ 48____No____ 63____Yes___ 65____NO___ 68___Yes____ I know that the ...
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7answers
332 views

Proving area under the integrals.

I have a question that I have been trying to solve that I am curious about. If you have a continuous function $f(x) = \frac1x$. How would you prove that ...
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1answer
38 views

Let f : [a,b] ---> R be a differentiable function…

I realize the hint tells me to use the mean value theorem, but I don't quite understand how to begin to relate it to this question? Any help is appreciated. http://i.imgur.com/HiT1QN5.png
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0answers
28 views

Find the values of $a$ and $b$ so that f will be continuous on $(-\infty, \infty)$

Find the values of $a$ and $b$ so that the function will be continuous on $(-\infty, \infty)$ $$ f(x) = \begin{cases} \dfrac{6}{x-4} & x<-2 \\ a-bx & -2\le x \le1 \\ \dfrac{x-1}{\sqrt ...
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1answer
16 views

Why is the phase shift -c/b instead of -c

In a function like $\sin(2x + 3)$ why is the phase shift $\frac{3}{2}$ units to the left instead of 3 units to the left
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1answer
31 views

A Question Regarding Remainder Theorem

What is the remainder when $x^3 + 3x^2 - x - 2$ is divided by $(x+3)(x+5)$? You have to solve this using the remainder theorem, which states: If $f(x)$ is divided by $(x-p)$, giving a quotient $g(x)$ ...
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1answer
35 views

Continuous functions unbounded on set

For Each of the sets construct a continuous function that is unbounded on the set. $\Bbb N$ $(2,3)$ $\left\{\frac 1 n \mid n \in \Bbb N\right\}$ $[0, \sqrt 2]\cap \Bbb Q$ ...
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1answer
30 views

Transform $\tan$ to be continuous between $0$ and $1$

I'm trying to create a $\tan$ function which has asymptotes between $0$ and $1.$ This is the closest I have gotten, but I can see that the asymptote is not actually at $1$ and when $x=0.5,\; y=0.02$. ...
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0answers
22 views

Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$

I'm trying to solve this problem: Get the critical points and find the máximum or mínimum of $f(x,y,z) = (x^{2} + 2y^{2} +1)\cos{z}$ First, I founded the gradient: $\nabla f(x,y,z)= (2x\cos{z}, ...
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0answers
32 views

Using calculus to find inverse functions

High schooler here. Last summer I taught myself a little bit of calculus, and I have been doodling about it. So I began writing some problems for myself, and one of them was this: Find the inverse ...
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1answer
35 views

Solving functional equation $f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n).$

Given natural number $n, k$ consider nondecreasing function $f:\mathbb{N}\cup {0} \to \mathbb{N}\cup {0}$ such that $$ f\left(\sum_{i=1}^n a_i^n\right)=\frac{1}{k} \sum_{i=1}^n f(a_i^n), $$ for ...
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1answer
37 views

Could you give an example of an injective function $f:\mathbb{Z_+}^n\rightarrow \mathbb{Z_+}$ for an integer $n$ s.t. $2\leq n$?

We know that both of the domain the the co-domain are countable sets, so there is a bijection between them, Is there any SIMPLE injection? Here is some injection which I thougt of, but It turns out ...
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1answer
98 views

$f(AB)=f(A)f(B)$, show that $f$ is or injective or zero

Let $f\in\mathcal{L}(\mathcal{M}_n(\mathbb{R}))$ such that: $\forall(A,B)\in\mathcal{M}_n(\mathbb{R}),f(AB)=f(A)f(B)$ How can I show that $f$ is or injective or the null function ? What I have ...
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0answers
25 views

Using function composition, show that $f(x)$ and $f^{-1}(x)$ are inverses of one another.

Consider the function $$f(x)=\frac{1}{3}x+2.$$ a) Find the inverse of $f(x)$ and name it $g(x)$. Show and explain your work. b) Use function composition to show that $f(x)$ and $g(x)$ ...
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1answer
32 views

Derivative Functions [closed]

Consider $f(x)= ax^2 + bx$ where $a$ and $b$ are real numbers. If $f(1)=-1$ and $f'(-1)=-7$, find the values of $a$ and $b$? I genuinely do not understand how to do it! Please help
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1answer
49 views

How many functions are there from 5 to 0?

I have just learned that from a set of $n$ element to a set of m elements, the number of functions is $m^n$. However, how about from $5$ to $0$? $5$ is a natural number which can also be considered as ...
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1answer
18 views

Discontinuities with an oscillating function in the denominator

The problem is to find the discontinuities in the following function \begin{equation} f(x) = \frac{4x+1}{5cos(\frac{x}{2})+1} \end{equation} I know the function will be discontinuous whenever ...
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2answers
56 views

Why isn't $f(x)=\sqrt{2-x}$ reflected across the y-axis?

If I try to graph this function, it does not appear to reflect across the y-axis when it comes time to do the reflection. Rather, it is reflected around the point where the function begins on the ...
2
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1answer
105 views

How do I find the PMF of X when X is the number of flips of a fair coin that are required to observe the same face on consecutive flips?

How do I find the PMF of $X$ when $X$ equals number of flips of a fair coin that are required to observe the same face on consecutive flips? The hint was to draw some sort of a tree diagram, but I'm ...
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0answers
28 views

Help with understand the growth order of functions

I am taking an Algorithms class and I understand everything that relates to the asymptotic growth and Order of growth for a given function (Theta, Omega, etc). However, I am having trouble in ...
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14 views

Empirical Intensity Function

I would like to ask for help determining what other ways are there to compute the "empirical intensity function" of a process. In essence, given that I observe the occurrences of an event in time ...
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0answers
10 views

Is it possible for the derivative of a multivariate function to be a function of lesser dimension?

Let's say I have some function $f$ such that $f'(a,b,c,d)$ exists for all $a$, $b$, $c$, and $d$, and that $f(a,b,c,d)$ is dependent upon $a$, $b$, $c$, and $d$. (That is, $f(a,b,c,d)$ can't be ...
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0answers
56 views

How do find out if a piecewise function has a maximum or minimum/how many?

Given a piecewise function with the domain $[0,2]$ and $f(x) = \begin{cases}2x^3 - x^2& \text{if } x \geq 1\\ \frac{x+1}{x-1}& \text{if } x<1\end{cases}$ How does one decide anything ...
5
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1answer
31 views

newbee question: How to create a function?

I wonder how to create a function based on the characteristics. suppose I have function $f$ and $g$ like this: $f(x,g(x,y,z)) = y$ $\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,,$ $g(x,f(x,z),a) = z$ With ...
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0answers
42 views

Proving inequality equation

Let $a_1, a_2,....,a_n$ be positive numbers such that $\sum_{i=1}^n a_i = 1$ Then for any vector $(x_1,x_2,...x_n) \ge (0,0,...,0)$ I want to show that $$x_1^{a_1}*x_2^{a_2}*...*x_n^{a_n} \le ...
0
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1answer
28 views

Piecewise function and maximum/min

$f$ is given by $$f(x) = \begin{cases} x^2 \cdot \sin(x) & \text{if }x\geq0\\ 1/x&\text{if } x < 0 \end{cases} $$ and if we look at the interval $[1,2]$ Given the above, how can we use ...