Tagged Questions

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

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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-...
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Domain of $f(x)=x^{\frac{1}{\log x}}$

What is the domain of $$f(x)=x^{\frac{1}{\log x}}$$ Since there is logarithm , the domain is $(0 \: \infty)$ But the book answer is $(0 \: \infty)-\{1\}$ but if $x=1$ $$f(x)=1^\infty=1$$ So is it ...
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homeomorphisms of the real line

Given a homeomorphism $h$ of the extended real line. Is it true that there exists an extension $\hat h$ of $h$, which is a Mobius transformation of a hyperbolic space $\mathbb{H}$? Any hints are ...
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How to approach general solutions to functional equations of multiple variables

I understand the concept of a function, broadly speaking, but when it comes down to solving general functional equations, I sometimes find it difficult to wrap my head around the problem at hand. For ...
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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 $f'(x)$ (...
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What is the simplest way to solve this primitive function?

I have some trouble with the rules to calculate the primitive function where $y = a^n$. For example: $$\int(4x^3(x^4+5)^5)dx$$ I want to break out the constant 4 to get: $4\int^\ (x^3(x^4+5)^5)dx$, ...
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Name for mappings where there is at least one y for every x

There are names for several properties of mappings from $x$ in $X$ to $y$ in $Y$. I think we say that a mapping from X to Y is (a)... Function: there is at most one $y$ for every $x$ Injective: ...
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Given the function $f:[0,1]→[0,1]$; $f(x)=x^2$, check which one(s) of the properties it has.

This homework is past due, but I am still fiddling trying to figure this out. question: I do not understand what the heck the notation of $f:[0,1] \to [0,1]$; means. I thought I did, but my ...
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What is the difference between a polynomial and a function or can they be used interchangebly?

I have been wondering over this basic question (seems rather trivial at first sight) for a long time- What is the difference between a polynomial and function? My confusion arises form the ...
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I can't find an appropriate piecewise function for this graph [closed]

On one of my piecewise questions I've split a graph into an exponential function, a cosine function and a parabolic function. I've done fine for exponential and parabola but I'm totally stuck on ...
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the function such $p|f(x)+f(y)$,then we have $p|x+y$.find $f(1)$

Question: Aussme that the function $\color{blue}{f:N^{+}\to N^{+}}$,foy any $x\neq y$,if $\color{red}{p|f(x)+f(y)}$,then we have $\color{blue}{p|x+y}$, find $\color{red}{f(1)}$ where $p$ is ...
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Trouble finding the inverse of $f(x) = x + \frac{1}{x}$ .

Let $f: \Bbb R - \{0\} \rightarrow \Bbb R \;\text{ given by } f(x) = x + \frac{1}{x} . \text{Find}$ $f(f^{-1}(\Bbb R))$ , $\Bbb R = \text{real numbers}$. For this problem I know one needs to ...
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Inverse Equation of the Given Equation

Having a bit of a problem getting the inverse of the following equation: $$f(x) = \sqrt{9-x^2}$$ I had an answer which was equal to $3-x$ but when I used sites like Mathway and Wolfram to check my ...
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Find the values of $b$ for which the equation $2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$ has only one solution

Find the values of 'b' for which the equation $$2\log_{\frac{1}{25}}(bx+28)=-\log_5(12-4x-x^2)$$ has only one solution. =$$-2/2\log_{5}(bx+28)=-\log_5(12-4x-x^2)$$ My try: After removing the ...
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Are functions $f(x) = \frac{x^2 + x }{x+1}$ and $g(x) = x$ equal? [closed]

Given $$f(x)= \dfrac{x^2 +x }{x+1} \qquad \qquad \qquad g(x) = x$$ Is it true that $f=g$?
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Question involving two functions f and g

f(x)= $\ x^2$ , $\ x > 2$  g(x)= $\ x^2$ , x $\in$ [0,4] Explain why f and g are different functions
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Notation of the square (or other power) of a function $f(x)$

How do you notate the square (or other power) of a function $f(x)$? Is it $f^2(x)$ (similar to $\sin^2(x)$ for example), $f(x)^2$ or do you have to use $(f(x))^2$? Thanks in advance.
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Can the derivative prove my function has only one root?

I have a function: $$f(x)=x-\ln(x^2+1)+2$$ I want to prove my function has exactly one root. If I differentiate: $$f'(x)=1-\frac{2x}{x^2+1}$$ I can see this value is positive for every $x$. Does this ...
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The strange “turning point” of $\arctan(x)/\arctan(\sqrt{x})$

After looking at an interesting graph: $$y=\frac{\arctan(x)}{\arctan(\sqrt x)}$$ There seemed to be a turning point around $(3{,}88;1{,}198)$ (https://www.desmos.com/calculator/58wloddve3) <- A ...
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The total number of solutions (real) of equation: $2^x+3^x+4^x-5^x=0 ?$ [duplicate]

The total number of solutions (real) of equation: $2^x+3^x+4^x-5^x=0 ?$ I have no idea how to solve this problem. Can someone point me in the right direction?
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If domain of $f(x)$ is $[-1,2]$ then what will be the domain of $f([x]-x^2+4)$ $?$

If domain of $f(x)$ is $[-1,2]$ then what will be the domain of $f([x]-x^2+4)$ $?$ Here $[.]$ is for greatest integer function. Attempt: since domain of $f(x)$ is $[-1,2]$ therefore for $f([x]-x^2+4)$...
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Value of $f^2(4)+g^2(4)$ [duplicate]

If $f(x)=g'(x),g(x)=-f'(x)$ for all real $x$ and $f(2)=4=f'(2)$ then value of $f^2(4)+g^2(4)$ is ? Now the above is true when we have a constant function with constant $0$. But then that would not ...
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Is this infinite series of continuous functions $f(x)=\sum_{n=1}^{\infty} \sin(\frac{x}{n^2})$ continuous?

The original question: Consider the function $$f(x)=\sum_{n=1}^{\infty} \sin\left(\frac{x}{n^2}\right).$$ Is $f$ a continuous function on $\mathbb{R}$ ? I know that the infinite sum of continuous ...