# Tagged Questions

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### how to find uniform continuity

I have some questions on continuity. What is the difference between continuous and uniformly continuous function? Please explain with this question. Find $f(x)=x^2$ is uniformly continous on ...
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### Heaviside Unit Step Function

Convert to heaviside function: $$f(t) = \begin{cases}e^t ,& 0 \leq t \leq 1 \\0 ,& t > 1\end{cases}$$ My attempt: $f(t) = U(t) e^t - U(t-1) e^t$ I think my solution is not right because ...
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### How to find an example

I want to find a function $f\in C^1([0,+\infty)\times\mathbb{R},\mathbb{R})$ such that $f(t,0)=0$ $f(t,u)\leq \alpha u+\beta$, $\alpha<\lambda_1,\beta\geq 0$ $f(t,u)\geq C_1 |u|^{\sigma}$ where ...
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### Composition of injections (proof)

I'm trying to prove that composition of injections is an injection. I want to know if this is a good proof: Composition of injections is an injection. Let $f:S_1\rightarrow S_2$ and ...
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### If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$.

If $f(g(x))=\sqrt {x^2-2x+8}$ and $f(x)=\sqrt x,$ find $g(x)$. There is no example like this in my math book.
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### problem on continuity [closed]

For $x>0$, let $[x]$ denote the largest integer less than or equal to $x$. Let $f:[0,\infty)\rightarrow\mathbb{R}$ be given by $f(x)=[x^2+[x^2]]\sin(2\pi x)$. Then $f$ is continuous at $2$ or ...
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### Sketch $y=2x^3/(x^2-2)$ [closed]

Sketch the curve $$y=\frac{2x^3}{x^2-2}.$$ Can someone answer this for me as basic as possible. Year 11 extension if possible. Thanks
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### Hypergeometric Distribution Function?

I'm looking for a function that I can use in excel to calculate the probabilities of having certain cards in an opening hand. For example a function that will calculate the probability to get AT ...
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### Surjectivity of composition

I know that this question has been posted few times, but I want to check MY proof, because this is my first time trying to prove anything in mathematics. (I'm afraid if I just copy paste their proofs ...
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### Getting to answer on difference quotient/function problem

Q: Find the difference quotient $\dfrac{f(x) - f(3)}{x - 3}$ for $f(x) = \dfrac{1}{x}$ Ans a: $\dfrac{1}{3x}$ Haven't been able to get to that answer. I got the bottom $3x$ right once but the top ...
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### Existence of injective function in a manifold with special atlas

I am trying do the following question: Let $M$ be a $n$-dimensional smooth manifold that admits an atlas with only two charts. Show that there exists an injective smooth map ...
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### Finding the range and domain of $f(x)=\tan (x)$

I am attempting to find the range and domain of $f(x)=\tan(x)$ and show why this is the case. I can seem to find the domain relatively well, however I run into problems with the range. Here's what I ...
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### Finding the range and domain of $h(x) = \sec (x)$

I am attempting to show how to find the range and domain of $h(x) = \sec (x)$. Here's my working so far. Consider $h(x) = \sec (x)$, which is defined as $h(x) = \sec (x)=\frac{1}{\cos(x)}$. We know ...
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### Functions and Relations - Help!

Given that : \begin{align} &f: D_1 \rightarrow \mathbb{R} \\ & g: D_2 \rightarrow \mathbb{R} \end{align} Find, $f + g : D_1 \cap D_2 \rightarrow \mathbb{R}$.
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### What is integral of $x^x$?

I have no idea on how to approach this problem. I tried solving it by taking logarithm and then evaluating, but that won't serve the purpose I guess. Can someone please help?
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### What is the linear analog of cusp? And difference between cusp and pole?

say some function has a singular line. Is that pole? If yes then what is the difference but cusp and pole besides the former is a point and the later is line?
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### Transformations order?

If I have the function $$f(x)$$ would I do e.g. a stretch scale factor $1/a$ parallel to the $x$-axis followed by a translation of $b$ units to the left like this $$stretch: f_1(x)=f(ax)$$ and then do ...
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### For which sets, $X$ the relation is a partial function

Given $T=\left\{\ \left<A,B\right> \in (P(X))^2 | A\subseteq B \right\}$ For which sets, $X$, the relation $(P(X))^2-T \cap (P(X))^2-T^{-1}$ is a partial function? Here's my solution: ...
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### Modulus function (working out coordinates)

Lets say you have $y = -|3x - 1|$ when working out where it cuts the axis, particularly the x-coordinate you do the following when $y = 0, 3x - 1 = 0$ therefore $x = 1/3$ the modulus and the ...
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So as the question says finding domain of- $f(x) = \log(\log_{|\sin x|}(x^2-8x+23)-\large\frac{3}{\log_{2}|\sin x|})$ $\large f(x)=\log(\log_{|\sin x|}(x^2-8x+23)-\large\frac{3}{\log_{2}|\sin ... 2answers 49 views ### Find the$f(x)$from the given information So tomorrow I tackled a maths test where I faced a question which was saying, Question: Let$f:R-\{0,1\}\rightarrow R$be a function satisfying the relation ... 3answers 60 views ### Find how far runners travel on a circular track (trig) -How far has each runner traveled after 8 seconds? Though I just had to convert the rad/sec to rev/sec to get yards then multiply that by 8 seconds, but that isnt correct. Find the angle θ, in ... 3answers 61 views ### How to make a cos function into a sin function I need to convert this equation into a sin function: f(x) = 12 cos(2x + 1) − 3 I know cos(x)= sin (pi/2 -x) but other than that I dont know how to solve this problem 1answer 29 views ### Domain of definition of the function I was going through some questions of Relations and Functions and now I am stuck to one. Question says Question: Domain of definition of the function $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ... 1answer 61 views ### A simple function equation in calculus-1 course Here is a homework question:$f^2(\ln x)-2xf(\ln x)+x^2\ln x=0,\ f(0)=0,\ f(x)=$？ I don't know how to solve it. Thanks! 3answers 72 views ### How to calculate the range of$x\sin\frac{1}{x}$? I want to find the range of$f(x)=x\sin\frac{1}{x}$. It is clearly that its upper boundary is $$\lim_{x\to\infty}x\sin\frac{1}{x}=1$$ but what is its lower boundary? I used software to obtain the ... 2answers 47 views ### Small question about limit if i have$\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)-a|u|^{\tau-2}u}{u}=0$how to prove that$\displaystyle \lim_{|u|\rightarrow 0}\frac{f(t,u)}{|u|^{\tau-2}u}=a$such that$\tau\in (1,2)$I ... 1answer 60 views ### Question about infinitely many times differentiable function. Could you please give me some hint how to solve this problem: Suppose$f(x)$is infinitely many times differentiable function on R,$f(0)=f'(0)=f''(0)=0$. Prove : for all$A>0$exists some ... 1answer 55 views ### Do you use degrees or radians for trig functions? I was just wondering if you use degrees or radians in trig functions. For example if I have a degree of 0.5 would I do: Sin(0.5) or would I have to convert that to radians? Or does it not matter ... 1answer 62 views ### Find angle in radians on a Ferris Wheel John has been hired to design an exciting carnival ride. Tiff, the carnival owner, has decided to create the world's greatest ferris wheel. Tiff isn't into math; she simply has a vision and has told ... 2answers 27 views ### Determining quadratic coefficients without function I was given the graph : and was asked to say whether the coefficients$(a,b,c)$of the function$ax^2+bx+c$for each of the 2 graphs was either positive or negative. We are supposed to find these ... 2answers 62 views ### Is this true about the inverse sine? It is known that$ \sin(-x)=-\sin x \ $. Bbut when we say: $$\arcsin(-x)=-\arcsin x$$ Is this true? Is it the same with the other trigonometric functions "inverse"? 2answers 39 views ### Find all holomorphic functions on$\mathbb{C}$, except for some singularities, such that$|f(z)|\leq C(|z|^{3/2}+|z-1|^{-3/2}), z\in\mathbb{C}-\{1\}$First I wrote the Laurent series of$f(z)$around$z=1$: $$f(z)=\sum_{n=-\infty}^{-1}c_n(z-1)^n+\sum_{n=0}^{\infty}c_n(z-1)^n.$$ Now if$|z|$becomes very large, the first sum with the negatives ... 1answer 17 views ### Finding rate in exponential decay Using the exponential decay eqution: I = Io * e^(-kx) -k = rate, x = time, Io = initial amount I was asked to find the rate (-k). We were given the following information, when x = 2 I = 12 and when ... 1answer 36 views ### Calculus-Tangent Line Find the cordinates of the point on the curve$f(x)=xe^{2x}+1$where the tangent of the tangent line is horizontal. I am not sure of what to do. 2answers 88 views ### Functional inequation on$\mathbb{R}$:$f(x+y^2)-f(x)\geq y$I have the following equation: $$f:\mathbb{R}\rightarrow\mathbb{R}$$ $$\forall (x,y)\in\mathbb{R}^2,\ f(x+y^2)-f(x)\geq y$$ f is not necessarily differentiable/continuous/... (In fact, we can prove ... 4answers 73 views ### Finding the inverse of$f(x)=|x|-2$How would I find the inverse of the function$f(x)=|x|-2$? I have swapped$x$and$y$, and tried to isolate$y$, reaching up to$x+2=|y|$Whenever I see absolute values, I always break the problem up ... 1answer 54 views ### Finding the equation for a sinusoidal cycle/function given points. We are given the population of a fictional animal at different years: $$\begin{array}{l|r} \textrm{Year} & \textrm{Population}\\\hline 1945 & 347,0000\\ 1955 & 76,000\\ 1965 & ... 1answer 37 views ### Find the linear-to-linear function whose graph passes through the given three points Find the linear-to-linear function whose graph passes through the points (1, 1), (4, 2) and (30, 3). So by using the$$f(x)=\frac{ax +b}{x+d}$$I got my final answer to be ... 1answer 33 views ### f(x) = e^{-{1\over x^2}}+\int_0^{\pi x\over2}(1+\sin t)^{1\over2}dt for x\in(0,\infty) Let$$f(x) = e^{-{1\over x^2}}+\int_0^{\pi x\over2}(1+\sin t)^{1\over2}dt$$for x\in(0,\infty) Then which of the following are true? (A) f′ exists and is continuous. (B) f′′ exists ... 0answers 26 views ### how to find the coefficient for a function to be continous at all x I'm having a problem solving this question, we have just learnt it at school today and this is my homework. Could you help me please? Find the values of a such that f is continous for all values of ... 1answer 48 views ### Let z=\ln \tan\frac xy. What is z_x and what is z_y? Let$$z=\ln \tan\frac xy.$$What is z_x and what is z_y? Thanks ahead:) What I have tried:$$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$... 1answer 38 views ### Let f_n: D \rightarrow \mathbb{R}: f_n(x) = g(x)^n, n≥1. Necessary and sufficient conditions such that f_n converges? The Assignment: Let D := [a,b] with a<b and g: D \rightarrow \mathbb{R} be continuous. Observe the sequence of functions f_n: D \rightarrow \mathbb{R}: f_n(x) = g(x)^n, n≥1. List and ... 1answer 58 views ### Given |f(x) - f(y)| \le \frac{1}{2}|x-y| what are the points of intersection of the graph of y = f(x) and the line y = x? Let f(x) be a real-valued function, defined for all real numbers x such that$$|f(x) - f(y)| \le \frac{1}{2}|x-y|$$for all x,y. Then the number of points of intersection of the graph of y = ... 2answers 33 views ### Analytical approach to a quadratics problem I'm a bit rusty on functions and this exercise got me thinking quite a bit. The function y=x is tangent to the graph of a certain g function in x=0. Function g can be defined as: ... 2answers 35 views ### Is it continuous at (0,0)?$$f(x,y)=\begin{cases} \frac{xy}{x^2+y^2}, \text{ if } x^2+y^2\neq 0 \\ 0, \text{ if } x^2+y^2=0 \end{cases}$$Is it continuous at$(0,0)$? 1answer 46 views ### Sketching the spectrum of a signal The figure below shows Fourier spectrum of a signal$g(t)$Sketch the spectrum of the signal$2g(t)\cos^2(100\pi t)$. Show value in sketch. 2answers 42 views ### Help finding this set Lets define the following: Let A be a set. A is innumerable if and only if there exists a bijective function from A to$\mathbb{N}$Proof that there exists an innumerable set$B \subseteq \mathcal P ...
Could you give me some hint how to solve this question: Suppose $f$ is a differentiable function for all $0<x<1$,$f(0)=1,f'(x)>0$ in the given interval. It is obvious that $f$ is continuous ...
Does anyone know how to calculate integral of $\sqrt{ 1-\cos (x)}$ ? I tried several methods resulting in $-2\sqrt2 \cos (x/2) + c$, but this is wrong in accordance with the text book, so i dont know ...