# Tagged Questions

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

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### Is there a function whose antiderivative can be found but whose derivative cannot?

Does a function, $f(x)$, exist such that $\int f(x) dx$ can be found but $f' (x)$ cannot be found in terms of elementary functions. For example, if $f(x)=e^{x^2}$, then the derivative is easily ...
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### Let $S=\{0,2,4,6,8\}$, $T=\{1,3,5,7\}$. Determine whether each of the following sets of ordered pairs is a function with domain $S$ and co-domain $T$.

Let $S=\{0,2,4,6,8\}$ and $T=\{1,3,5,7\}$. Determine whether each of the following sets of ordered pairs is a function with domain $S$ and co-domain $T$. $\{(6,3),(2,1),(0,3),(8,7),(4,5)\}$ TRUE ...
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### Maximizing $f(0)$ given that $f(3)=5$ and $f'(x)\ge1$ [on hold]

Let there be $$f:(-1,4)→ R$$ $$\text{differentiable on} (-1,4) , f(3)=5 , f'(x)≥-1$$ $$\text{which is the maximum value of}$$$$f(0)$$
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### Simple formula difficult solution

I've thinking a lot about it, but is there a simple way to get $\frac{A}{C}$ from $X = \frac{A + B}{C + D}$ where it does not depend on A and C anymore? This seems so easy but it's quite hard for ...
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### Greatest Integer Function and Limits - Is GIF of $\sin x/x$ equals to $0$?

Okay, so I read this somewhere that, $$\lim_{x \to 0^+} \left[ \frac{\sin x}{x} \right] = 0$$ Where, [] denotes the greatest integer function. But, on the other hand, this is also true, ...
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### Functions - Find number of positive integral values of 'x' which satisfy an inequality

Let $f(x) = 30 - 2x - x^3$, then find the number of positive integral values of 'x' which satisfies $f(f(f(x))) > f(f(-x))$. The first thing that I saw in the above question was that the ...
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### Find all the angles $v$ between $-\pi$ and $\pi$

Find all the angles $v$ between $-\pi$ and $\pi$ such that $$-\sin(v)+ \sqrt3 \cos(v) = \sqrt2$$ The answer has to be in the form of: $\pi/2$ (it must include $\pi$) I have tried squaring but I get ...
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### Need know all ways to show function is continuous, convergent and differentiable [on hold]

Please tell me all ways to show / proof that a function is continuous, convergent and differentiable. continuous: show that function is differentiable if yes then it is continuous also convergent: ...
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### Limit of functions - always for both sides (+-) necessary?

I'm very confused when I read some pages on the internet about limits (for functions). Let's say I got any function f(x) given and someone tells me to find the limit (towards 3 or $\infty$ or ...
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### Correct Order of Applying Graphical Transformation with Absolute Value

I was going through this website, reading about transformations of graph when $| |$ is applied to various parts of a given function, $y=f(x)$. Going through the fourth example of the page, I came ...
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### Finding a delta for the greatest integer function given an epsilon = 1/2

I'm having trouble with the following problem. Given the standard greatest integer function $\lfloor x \rfloor = int(x)$ where $\lfloor x \rfloor$ returns the greatest integer less than or equal to ...
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### Determine the range of f(x)=(sinx)/x

I am having trouble understanding the solution to this question. ''Determine the range of the following function: $f(x)$ = $(1$ $if$ $x=0)$ or (${\sin x\over x}$ if $x$$\neq$$0$) where the domain ...
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### Improving inequality $(\int X(x) Y(x) \,dx) \leq (\int |X(y)| \,dy) Y_{\max}$

Want to improve the following inequality: $(\int X(x) Y(x) \,dx) \leq (\int |X(y)| \,dy) Y_{\max}$ Looking to replace $Y_{\max}$ with something that will give a tighter bound. Everything else needs ...
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Define $$x'=\text{argmin}_{x_1}f(x_1,\lambda),$$ where $f$ is a strictly convex function on $x_1$ and $\lambda$. I would like to ask if there is any theorem about the continuity of $x'$ w.r.t $\... 1answer 17 views ### tangents conditions what are the condition for a tangent to be exist . Is it necessary for the function to be continous. but it is necessary to be continous for a function to be differentiable at that point . can ... 2answers 2k views ### If$f(x)$has a vertical asymptote, does$f'(x)$have one too? So here is what I understand: If$f(x)$is increasing/decreasing, then its derivative$f'(x)$is positive/negative and... If$f(x)$is increasing/decreasing, then the derivative of$f'(x)$(... 3answers 63 views ### How to solve$x_{100}$with$x_{n+1}=\frac{2x_n}{2+x_n}+1$and$x_{1}=3$How to solve$x_{100}$with$x_{n+1}=\frac{2x_n}{2+x_n}+1$and$x_{1}=3$? Can anybody shed light on this? regards. 0answers 20 views ### Find a hypergeometric formula embracing three specific cases For a parameter value$a=\frac{1}{4}$, I have the result Q(k,\frac{1}{4})=\frac{2^{-2 k-\frac{19}{4}} \Gamma \left(2 k+\frac{13}{4}\right) \, _3F_2\left(1,k+\frac{13}{8},k+\frac{17}... 0answers 24 views ### I want to show that the function space$C_0(X)$is Banach [duplicate] I'm reading some papers but I encountered a problem that "$C_0(X)is Banach space". Here C_0(X):= \{ f: X\to \mathbb{C}: f \text{ is continuous and } \forall \epsilon>0, \exists K(\text{compact}... 0answers 10 views ### Determine the operation based on the conditions given below \begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions f, g, h are defined for all a,b,c,d\in\mathbb R. For instance: h can be Division; a, b, ... 0answers 30 views ### Solve for the conditions given below [closed] \begin{align} f(c, d)&= a;\\ g(c, d)&= b;\\ h(a, b, c)&= d. \end{align} The functions f, g, h are defined for all a,b,c,d\in\mathbb R. For instance: h can be Division; a, b, ... 4answers 148 views ### inequality \sqrt{\cos x}>\cos(\sin x) for x\in(0,\frac{\pi}{4}) How can I prove the inequality \sqrt{\cos x}>\cos(\sin x) for x\in(0,\frac{\pi}{4}) ? The derivative of f(x):=\sqrt{\cos x}-\cos(\sin x) is very unpleasant, so the standard method is ... 1answer 61 views ### Define an \mathbb{N} to \mathbb{N} function that is Hi I'm preparing for an exam and was going through exercises on functions. I stumbled upon this question and didn't know how to answer it. Give an \mathbb{N} to \mathbb{N} function that is one-... 1answer 22 views ### How to know describe the set of levels for functions f(x,y)=c when c varies Hey im having quite troubles trying to understand how to describes the set of levels in functions. In this problem any ideas?f(x,y)=x^2+y^2+1$$0answers 25 views ### A problem about a continuous iterated function [duplicate] Let f:\mathbb {R} \rightarrow \mathbb { R } be a continuous function such that f\circ f \circ f=\text{id}_\mathbb{R} . Show that f=\text{id}_\mathbb{R}. Is there any hint to prove this? ... 1answer 19 views ### increasing and one one function If we are given a function f(x)=x^3$$+$3x$ for all x belong to real number . Now as the derivative of function is always positive so the function should be increasing function and if it is always ...
The function f is continuous and has the property $f(f(x)) = 1 - x$ for all x in $[0, 1]$, and $J = \int_{0}^{1} f(x) dx$, then find $f(\frac{1}{4}) + f(\frac{3}{4})$ & the value of J. I ...