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

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1answer
125 views

Find the smallest root of the function $e^{-x} = \sin (x)$

I have the following problem: Find the smallest root of the function $e^{-x} = \sin (x)$ and focus the root with Newton's method to $8$ decimal accuracy. Any suggestions?
1
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2answers
83 views

What conclusion can we draw?

Let $f \colon Z \to \{-1, 1\}$, where $Z$ denotes the set of integers, be defined by $ f(n) = 1$ if $n$ is even and $f(n) = -1$ if $n$ is odd. Then we can easily show that $f(m+n) = f(m) \cdot f(n)$ ...
5
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1answer
1k views

If $|A|=30$ and $|B|=20$, find the number of surjective functions $f:A \to B$.

Let there be: $|A|=n$ and $|B|=m$ if $m>n$ then there are $$m(m-1)\cdots(m-n+1)$$ injective functions, so in this case we have $|A|=30$ and $|B|=20$ that means $m<n$ so there exists a ...
0
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1answer
29 views

Rewriting function in Warren textbook

Not exactly sure how to phrase the title, because the question I have is rather specific and I think requires a bit of background. The question is about the rewriting of a function that appears in ...
0
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1answer
367 views

Show $g(x)=\sqrt{x}$ is continuous at x=4

I just really need to make sure that I am understanding the process for doing these. Scratch work: We have $|\sqrt{x}-\sqrt{4}| = |\sqrt{x}-2|= |\frac{x-4}{\sqrt{x+2}}|= \frac{|x-4|}{|\sqrt{x}+2|}$. ...
0
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1answer
118 views

Prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2.

I am looking to prove that the function $f(x)=\frac{1}{x}$ is continuous at the point x=2. So we nee that given any $\epsilon>0,\ \exists\delta>0$ so that $|f(x)-f(2)|<\epsilon\\$ whenever ...
2
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0answers
86 views

Is there any intuition why this relation holds?

If $$D=\frac{d}{dx}$$ then we have $$\exp(D)f(x)=f(x+1)$$ What does it mean? Is there any intuition behind it?
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1answer
211 views

Ratio of convex functions with dominating derivatives is convex?

Let $f,g:\mathbb [0,\infty)\rightarrow (0,\infty)$ satisfy $f^{(n)}(x)\geq g^{(n)}(x)>0$ for all $n=0,1,2,\ldots$ and $x\in [0,\infty)$. In particular, $f\geq g> 0$ are increasing and convex ...
1
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1answer
392 views

Check for monotonicity, convexity

I have the following function: $$ f(x) =\sin(x) - \sqrt{3} * \cos(x) $$ and $$ I = [-\pi , +\pi] $$ I should check this function for monotonicity and convexity. So I drawed the $\sin$ and $\cos$ on ...
0
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1answer
60 views

Function to increase X by Y while keeping its negative or positive value

I hope my question is not too dumb. I need to make a function that adds Y to X towards its sign. I know I'm probably not very clear, so here is an example. If X = -3 and Y is 2, the results is -5. ...
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4answers
146 views

Inverse Function of $ f(x) = \frac{1-x^3}{x^3} $

I need to show that these function has a continuous inverse function and find this inverse function. $$ f(x) = \frac{1-x^3}{x^3} $$ Defined on $ (1,\infty) $ I think I need to check for ...
0
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2answers
36 views

Prove the lines given by the functions

There is the second problem of the day that I have been stuck on for quite some time, and I am having trouble examining how to evaluate this equation to simple form. Prove the lines given by the ...
1
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3answers
46 views

Prove directly that for $x,y \ge 0$

The question that I am stuck on is as follows: Prove directly that for $x,y \ge 0$ $\sqrt{xy}\le (x+y)/2$. When does equality hold? I have been working at it for almost 20 minutes. Can ...
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3answers
1k views

Teaching the concept of a function.

I am doing a class for at risk high school math students on the concept of a function. I have seen all the Internet lesson plans and different differentiated instruction plans. The idea of a ...
0
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2answers
70 views

About preimage of a function.

I'm trying to show that $f(f^{-1}(E))= E$ is not necessarily true. I want to use a non-injective function as an example, but what I want to know is if I can consider $E$ to be a set which is not a ...
0
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4answers
104 views

bijection in $\mathcal P \left({S}\right)$ and $\mathcal P \left({S}\right) \times \mathcal P \left({S}\right)$

given that ${S}$ is countably infinite set. is there any bijection exist between $\mathcal P \left({S}\right)$ and $\mathcal P \left({S}\right) \times \mathcal P \left({S}\right)$. Here $\mathcal ...
3
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3answers
276 views

A simple but weird functional equation

Let $f$ be a function $f:\mathbb R\to\mathbb R$. Find all functions $f$ that satisfy: $$f(x^2+x+3)+2f(x^2-3x+5)=x^2-x+ \frac{18}{4} + \frac{111}{444} + \frac{222}{333}$$ Maybe the question is ...
0
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1answer
49 views

Complex Analysis

Determine the derivative of the following functions and state where they are analytic. $$f(z) = \log(z^3) \quad \Rightarrow f'(z) = \frac{3z^2}{z^3} = \frac{3}{z}$$ Hence, this function is analytic ...
0
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1answer
71 views

Bijection between $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R}$

How can we define bijection in between $\mathbb{R} \times \mathbb{R}$ and $\mathbb{R}$? Even giving a injection from $\mathbb{R} \times \mathbb{R}$ to $\mathbb{R}$ and vice-versa will work.
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2answers
180 views

If $f(x) = \frac{x}{\sqrt{x^2 + 1}}$, what is $\underbrace{f(f(f( \dots f}_{2013}(x) \dots )))$?

Given the function $$f(x) = \frac{x}{\sqrt{x^2 + 1}},$$ I need to evaluate the iterated (nested) function $$\underbrace{f(f(f( \dots f}_{2013}(x) \dots ))).$$ I believe the alternative notation ...
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2answers
424 views

Prove the Existence of an Inverse Function

We're asked to prove formally that if $f: X \rightarrow Y$ is one-to-one, then there is a $f^{-1}: f(X) \rightarrow X$ such that $(f^{-1} \circ f)(x) = x$. Let $f = \{ (x,y) | x \in X, y = f(x) \}$ ...
2
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1answer
681 views

How to prove a function is periodic?

$$f(x)=\begin{cases}1&\text{if }2n-1<x<2n,\\0&\text{if }2n<x<2n+1. \end{cases} $$ Is this function is periodic or not? How can I prove it?
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3answers
122 views

Self multiplication of a CDF degenerates into a Dirac Delta?

Because of some work I need to do, I bumped into this problem. A CDF $F(x):\mathbb{R} \mapsto [0,1]$ of some random variable is raised to the n-th power. The function is continuous all over its domain ...
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0answers
78 views

Picking an arbitrary element from an equivalence class

I want to define a function $f$ over a quotient set $E/R$. The trouble is that I want to say $f([M]_R) = g(M')$ for some $M'\in[M]_R$ and $g$ a certain previously well-defined function. Any member ...
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2answers
305 views

Proving that $A\subseteq f^{-1}(f(A))$ and that if $f$ is injective $A=f^{-1}(f(A))$.

this is an intro to analysis problem on Set Theory, Relations and Functions: Let $f:S\to T$ be a function and $A\subseteq S$. Prove that $A\subseteq f^{-1}(f(A))$ and that $A=f^{-1}(f(A))$ if $f$ is ...
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1answer
65 views

Trajectory of circumference [circle] rolling down any given curve

How should i go about describing mathematically the path traced by the center of a circumference [circle] rolling down (or up) any given curve described by $y = f(x)$? The solution for a linear ...
2
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1answer
144 views

Proving the inequality $\frac{\tan{x}}{x} > \frac{x}{\sin{x}}$ [duplicate]

I'm trying to prove this inequality: $$\frac{\tan{x}}{x} > \frac{x}{\sin{x}} $$ for all $x$ in $(0,\frac{\pi}{2})$. I tried analyzing the derivates, but that's just making it more complicated. Any ...
2
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0answers
88 views

On finding new mathematical constants and patterns

How does one go about finding mathematical constants and patterns... I enjoy playing around on my graphing calculator, typing random functions in trying to find patterns and such and sometimes I'll ...
0
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5answers
148 views

Function Inversion

I have this equation: $f(x) = x + 3$ I need to invert it. Could somebody do it for me and explain how it's done? Thank you!
1
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2answers
132 views

Preimage of a function

The only way to get better at this sort of thing is to practice, and now I'm also trying to ask myself (and try to answer) more conceptual questions. If a circle with radius $r$ is given in ...
0
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5answers
108 views

How do I solve this confusing Function problem?

So I'm not sure what this kind of problem is called, but I am in my fist semester of Calculus and I am really confused. Here is the actual problem: A function f is given that satisfies ...
3
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1answer
204 views

Polygon sine waves

So I came across this picture on Google+ and I wanted to understand further. I created an equation for the second wave, the one with the square. Here it is: $$y=\frac{\sin x}{\cos(\min(x \mod \pi/2, ...
0
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1answer
92 views

Can surjective functions map an element from the domain…

Can surjective functions map an element from the domain to two distinct elements in the codomain? I know that each element in the codomain must have at least one corresponding element in the domain. ...
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1answer
34 views

Quantifying “weight” or “control” of a variable to the value of a function?

Say I have an equation $$x = f(x)$$ I know that here, the independent variable $x$ controls 100% of the variability of the value of the function. It is the only "knob" that I need to turn to ...
1
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1answer
116 views

$f(x)=f(x^2+ 1/4)$ , $f$ is continuous from $\mathbb{R}$ to $\mathbb{R}$

Find all continous functions $f\colon\mathbb{R}\rightarrow\mathbb{R}$ satisfying $f(x)=f(x^2+ 1/4)$ What I've tried so far: suppose that $f$ is one-one thus $x=x^2+1/4$ ... $x=1/2$ then ...
0
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1answer
173 views

Converting to polynomial form

Is it true that if we have an equation of the form $g(a)=0$ ( $g:R→R$) then it is possible, with certain manipulations, to convert this equation into the form $P(a)=0$, where P is a polynomial?
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1answer
38 views

Total angle within a closed surface…

So I know that the degrees of a triangle add up to 180 degrees. This seems to make sense, on some intuitive level. Now, if you have a closed object constructed of four lines (perhaps most simply a ...
0
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3answers
115 views

Find inverse of this function…

$f(x) = \sin(x)-\cos(x)$ I got to this point: $$f^2(x)=1-\sin(2x)$$ But I have no idea what to do next. Please help me, give me a hint :D
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3answers
83 views

Minimum and Maximum .

In a book it has been written that "Taking the minimum and maximum of each term we see that on $[0,1]$ the function, $y=x^3-7x^2+1$" , is bounded below by c$=0-7+1=-6$ and above by d$=1+0+1=2$. I ...
1
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1answer
172 views

How to find x-intercepts of the graph of $f$

Given the following function $$f(x) = \frac {\sin (4x)}{2x}$$ How to find $x$-intercepts of the graph of $f$ on the interval $0 \le x \le \pi$?
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2answers
137 views

Let $f$ be a homomorphism from the reals under addition to the nonzero complex numbers under multiplication. Find the image of $f$.

If $f$ is as given in the problem statement, then how do I determine its image? My book says that the image of $f$ = {$z$ in the nonzero complex numbers under multiplication such that $z=f(x)$ for ...
2
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5answers
188 views

If a function $f$ satisfies $f(2x+3)=x^2$, how to find $f(0)$?

If a function satisfies $f(2x+3)=x^2$, what is $f(0)$? Explain how you figured it out, please.
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2answers
637 views

Evaluating $f(x)$ for values of $x$ that approach $0$

(a) By graphing the function $$f(x) = \frac{\cos 2x − cos x}{x^2}$$ and zooming in toward the point where the graph crosses the y-axis, estimate the value of $\lim_{x \to 0} f(x)$. I find that the ...
0
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2answers
40 views

Showing that $f: \Bbb Z \to \Bbb Z_n$ with $f(x)=[x]$ for each $x$ in $\Bbb Z$ is onto.

These easy concepts are evading me, and I'm struggling with function analysis. How do I show that $f: \Bbb Z \to \Bbb Z_n$ with $f(x)=[x]$ for each $x$ in $\Bbb Z$ is onto? This is not a homework ...
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2answers
55 views

General form of functions

Is there a general form for functions? For example if the function is a polynomial, the general form is well-known. But is there a general one, covering every possible function?
0
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1answer
101 views

How can I calculate the derivative of the derivative using the tangent line?

How can I calculate the derivative of the derivative of a function $ f(x) $ using the tangent line of a point from that function $ f(x) $ ?
7
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1answer
1k views

Prove that there is a bijection between the set of all subsets of $X$, $P(X)$, and the set of functions from $X$ to $\{0,1\}$.

Given any set $X$, let $P(X)$ be the set of all subsets of $X$, and let $\{0,1\}^X$ be the set of all functions $X \rightarrow \{0,1\}$. Construct a bijection (and its inverse) between P(X) and ...
1
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1answer
41 views

Possible for this function to be discontinuous?

Let $g:[0,T]\times \Omega \to \mathbb{R}$ where $\Omega$ is spatial domain, and $f:\mathbb{R} \to \mathbb{R}$. Is it possible for $$\frac{d}{dt}f(g(x,t))=f'(g(x,t))g'(x,t)$$ to be continuous in $t$, ...
0
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1answer
78 views

How is this $f(n) = n^n$ function called?

If $n^2$ is called quadratic function,$n^3$ is called cubic function then what is name for $n^n$ ? exponential
0
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1answer
111 views

Locus question?

Just wondering how you would solve this: "Find the locus of a point $P(x,y)$ which moves such that its distance from the $x$-axis is always one more unit than its distance from the $y$-axis." Thanks ...