0
votes
2answers
46 views

Geometric meaning of results obtained in (a) and (b)

The task: Plot the function $\sqrt{1-x^2}$. What does it look like? What is the geometric meaning of the results you obtained in (a) and (b)? Can anybody help me with geometric mean? I can't ...
0
votes
2answers
58 views

Does the definition range remains the same?

In solving this inequality (transcribed from here) $$\left(\frac23\right)^{\log_{0.5}(x^2+4x+4)}<\left(\frac94\right)^{\log_2(x^2-3x-10)}$$ we eventually reach the point where $ ...
0
votes
1answer
30 views

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 ...
2
votes
4answers
34 views

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 ...
0
votes
0answers
47 views

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 ...
2
votes
2answers
34 views

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 ...
1
vote
2answers
49 views

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.
-2
votes
3answers
60 views

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 ...
1
vote
2answers
62 views

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
0
votes
1answer
11 views

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 ...
0
votes
1answer
31 views

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 ...
0
votes
2answers
21 views

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 ...
1
vote
1answer
39 views

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 ...
1
vote
5answers
54 views

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 ...
0
votes
1answer
34 views

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 ...
0
votes
1answer
25 views

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} $.
1
vote
1answer
106 views

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?
0
votes
1answer
19 views

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?
-1
votes
0answers
18 views

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 ...
0
votes
0answers
11 views

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: ...
0
votes
1answer
17 views

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 ...
0
votes
2answers
55 views

Find domain, $f(x) =\log(\log_{|\sin x|}(x^2-8x+23)-\large\frac{3}{\log_{2}|\sin x|})$

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 ...
1
vote
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 ...
0
votes
3answers
61 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 ...
0
votes
3answers
63 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
1
vote
1answer
30 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 $$f(x)=\frac{9}{9-x^2}+\log_{10}(x^3-x)$$ ...
1
vote
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!
5
votes
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 ...
0
votes
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 ...
0
votes
1answer
62 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 ...
2
votes
1answer
61 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 ...
0
votes
1answer
64 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 ...
0
votes
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 ...
0
votes
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"?
2
votes
2answers
40 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 ...
0
votes
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 ...
0
votes
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.
2
votes
2answers
89 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 ...
2
votes
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 ...
3
votes
1answer
55 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 & ...
1
vote
1answer
39 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 ...
1
vote
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 ...
0
votes
2answers
34 views

Proving $f(x) = x^2 + 1$ is surjective

Let $f(x) = x^2 + 1$, where $x$ is a real number. Prove that $f$ maps $ \mathbb R$ onto $[1, \infty)$. We must show that if $y \in Y$, then there exists an $x$ such that $f(x) = y$. I am tempted ...
0
votes
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 ...
2
votes
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: $$z_x=\frac{1}{\tan \frac xy} \frac{1}{1+(\frac xy)^2} \frac 1y=\frac {y}{\tan \frac xy (x^2+y^2)}$$ ...
2
votes
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 ...
2
votes
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 = ...
1
vote
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: ...
-1
votes
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 $(0,0)$?
0
votes
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.