# Tagged Questions

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

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### How to transform function values to specific interval

I'm doing a project at university about scientific computing and I'm stuck. As in: I seem to lack quite a bit of mathematical background for this project. The program has as input an array of $x$ ...
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### Constant Moving Speed

I've made this graph: https://www.desmos.com/calculator/czk3ylyokj As you can see, the purple point is slowing down as it approaces the extreme point. How can I make this point move with constant ...
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### Find values of a and b if the function is continuous

First I took left hand limit and right hand limit but I am not sure how to simplify the equations and remove the 0/0 form . Looking for your help desperately. Have an exam tomorrow help is appreciated ...
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### English for “prolongement” oder “Fortsetzung”?

I'm sorry if that's not the right place to ask for, wikipedia failed to give me the correct word... What's the english for a function that is defined on a larger domain than the original function and ...
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### $F(x)+G(y)= e^{x+y}?$

Are there functions $F(x)$, $G(y)$, such that $F(x)+G(y)=e^{x+y}$ , where $x,y$ are real numbers? I have been trying all elementary functions, and have no clues on what else I could do.
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### Mean Value Theorem Inequality Contradiction(?)

I am trying to show: $e^x > 1+x+\frac{x^2}{2}$ for $x>0$ using Mean Value Theorem (MVT). My method is as follow: consider the function $f(x)=e^x - (1+x)$. We know $f'(x) = e^x -1 > 0$...
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### Unique solution of nolinear equation set

\left\{ \begin{aligned} f_1(x_1,x_2...x_n)=0 \\ f_2(x_1,x_2...x_n)=0 \\ \vdots \\ f_n(x_1,x_2...x_n)=0 \end{aligned} \right. $f_i\in C^\infty(R^n)$,what is the condition that make the equation ...
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### Show that the metric space C[a,b] is complete. [duplicate]

Prove that the metric space $C[a,b]$ is complete. Where $C[a,b]$ is the collection of continuous $f:[a,b] → R$ and $||f|| = sup_{x \in [a,b]} |f(x)|$, such that $\rho (f,g) = ||f - g||$ is a metric ...
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### How can I prove the inverse of a function is odd?

Given function $f$, where $A⊆ \mathbb{R}$ is a symmetric domain with respect to 0,$\;\; f:A \rightarrow\mathbb{R}$ and $f$ is an odd one-to-one function, I need to prove $f^{-1}$ is odd. I was ...
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### Suppose that $f : \mathbb{R}\rightarrow \mathbb{R}$ is continuous and that $f(x)\in \mathbb{Q}$ for all $x \in \mathbb{R}$. Prove $f$ is constant. [duplicate]

I am really stuck on this proving this statement, so could someone please help me get through this. Thank You. P.S. This can be proved through elementary analysis results instead of going into ...
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### How do I prove that $\arccos(x) + \arccos(-x)=\pi$ when $x \in [-1,1]$? [closed]

Prove that $\arccos x + \arccos(-x) = \pi$ when $x \in [-1,1]$. How do I prove this? Where should I begin and what should I consider?
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### Proving trigonometric identity $1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$

$$1+\cot x\tan y=\frac{\sin(x+y)}{\sin x\cos y}$$ I have worked through most of this question, and I believe I am so close to finding the answer, but I have run into some issues where I am not sure ...
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### limits involving a piecewise function, Prove that if $c \ne 2$, then f does not have a limit at $x = c$.

$$f(x) = \begin{cases} (x-2)^3 & \text{if x is rational } \\ (2-x) & \text{if x is irrational } \end{cases}$$ (i) Prove that if $c \ne 2$, then f does not have a limit at $x = c$. (ii) ...
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### Prove the trigonometric Identity involving secant

The question I am currently working on is: $\sec^2x-2\sec x\ \cos x+\cos^2x=\tan^2x-\sin^2x$. Okay, judging by the expression here, I am going to need to work with the left side of the equation first,...
### $\sin x \sin(\pi/2-x)=\sin x \cos x$?
During a question involving proving a trigonometric identity, I was given help in which one of the lines showed that $\sin x \sin(\pi/2-x)$ equals $\sin x \cos x$? Could anyone please explain to me/...
$\cot x=\sin x \sin(\pi/2 -x) + \cos^2x \cot x$ I'm having difficulty with figuring out how to prove trigonometric identities. I know that in order to do these you need to use the trig ratios ...