Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
2,756
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Find all $f:N_0\to N_0$ which obey the functional equation $2f(m^2+n^2)=f(m)^2+f(n)^2$ for all non-negative integers $m,n$
Find all $f:\Bbb{N_0}\to \Bbb{N_0}$ which obey the functional equation $$2f(m^2+n^2)=f(m)^2+f(n)^2$$ for all non-negative integers $m,n$.
My attempt
Putting $m=n=0$, we get $f(0)=0$ or $1$.
Case 1: $...
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votes
1
answer
94
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How total derivative of a function works and what is the derivation of the formula?
If f is a function of $(x,y,z)$ then the total derivative is
$$d f = \frac { \partial f } { \partial x } d x + \frac { \partial f } { \partial y } d y + \frac { \partial f } { \partial z } d z$$
but ...
- 132
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2
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Is this function injective or surjective: $g: \mathbb{N} \rightarrow \mathbb{N}, n \mapsto 2n^{3}-1$
Is this function injective or surjective: $g: \mathbb{N} \rightarrow
\mathbb{N}, n \mapsto 2n^{3}-1$
I don't know how I can say this. I have to find some values for which we see it cannot be ...
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1
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What's the image of the function $f(x)=(3x+2^{v_2(x)})$ on the Prufer 2-group?
What's the image of the function $f_r(x)=\left(\dfrac{3x}{2^r}+2^{v_2\left(\tfrac{x}{2^r}\right)}\right)$ on the Prufer 2-group $\mathbb{Z}_2(2^{\infty})$?
For each case of $r=0, 1,\text{ or }2$?
To ...
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1
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Proving that a real function is bounded [closed]
Let $y$ be a real function and $c>0$. If $y$ satisfies the following inequality
\begin{equation}
y^{\prime}(t)+y(t)\leq y^{\frac{3}{2}}(t)+c
\end{equation}
How can I prove that $y$ is bounded.
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Consider the function $f:\Bbb Q\to \Bbb Q$.Prove that $f$ is not monotonic,find it's range... [closed]
Consider the function $f:\Bbb Q\to \Bbb Q $.For every $m\in \Bbb Z,n\in \Bbb N$ with the condition $\gcd(m,n)=1$ we have: $f(\frac mn)=\frac{m}{n+1}$.Now:
a)Prove that $f$ is not monotonic(...
- 3,445
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1
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193
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Set theory proof, $\exists g(g\colon A\twoheadrightarrow C)$ or $\exists h(h\colon C\hookrightarrow B)$ [closed]
Let $A$, $B$ and $C$ be such that $|A \cup B|=|C \times C|$. Prove that $\exists g(g\colon A\twoheadrightarrow C)\lor \exists h(h\colon C\hookrightarrow B)$ ($h$ is injection and $g$ surjection).
- 123
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1
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602
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Calculation of average of sine wave based on area under the curve and amplitude [closed]
Is it possible to derive analytically the mean value of the sine wave function (expressed as $f(x) = mean + A*\sin(x)$) based on known area under the curve and amplitude ($A$) (for illustration of ...
-1
votes
1
answer
73
views
Is the formula $\exists x [f(x) = a]$ logically valid?
Since the variable $x$ belongs to the universal set, then we could say that $f^{-1}(a)$ is one of the values we could assign to $x$. Therefore, $\exists x [(f(x) = a]$ is equivalent to $f(x_0)=a \lor ...
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1
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What is a continuous transfer function?
What is continuous transfer and what does a continuous transfer function (like the CLR from SciLab) do? Why is it useful/where is it used?
- 87
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1
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129
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Existence of additive non-linear function [closed]
The following question should have a positive answer: it is taken from Example 1.11 of the book "Positive Operators" by Aliprantis and Burkinshaw.
Question: Does there exist an additive function $\...
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4
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Solving a problem using the definition of limit [closed]
How can I solve this using the definition of limit?
Prove using the definition of limit that:
$$\lim_{x\to 1} (x²-4x)=-3$$
How can I approach this?
EDIT: OH my god! Thanks @adam!
Maybe you can ...
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1
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Question about writing cyclometric function in function of $x$
I have an excercise about cyclometric functions and I'm stuck right now:
$\cot(2\operatorname{arcsec}x)$
Let $ y=\operatorname{arcsec}x \Leftrightarrow \sec y=x$ then $$\cot 2y=\frac {\cos2y}{\sin2y}...
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1
answer
82
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What this means,f(x)=y in case of surjective function? [closed]
$x$ means input, $y$ means output so $$f(x)=y$$ means any input that goes into this function give the $y$ that means output, but how this can prove a function is surjective?
- 193
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4
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Calculate function: $\int_{a}^{b} \left(f{(x)}\right)dx=c$
Is there a way to find the function $f{(x)}$ for a given value of $a,b,c$?
$$\int_{a}^{b} \left(f{(x)}\right)dx=c$$
For example:
$a=0,b=1,c=\frac{1}{3}$ we get:
$$\int_{0}^{1} \left(f{(x)}\right)...
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1
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How to create a function f(x,y) = z where an increase in |x-y| corresponds with an increase z and |y| corresponds with decrease in z [closed]
I'm trying to figure out a function that does the following:
Increases with a (that is, the absolute value of the magnitude between ...
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3
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Problem on theory of equations.
Let $P(x) := \prod_{k = 1}^{50} (x - k)$ and $Q(x) := \prod_{k = 1}^{50} (x + k)$.
If $P(x) Q(x) = \sum_{k = 0}^{100} a_k x^k$, find $a_{100} - a_{99} - a_{98} - a_{97}$.
The correct answer is ...
user411373
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1
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Do these points trace out a function? $ P(2^{2^s},2^{2^{-s}}) $
Do these points trace out a function? What is the functional equation?
$$ P(2^{2^s},2^{2^{-s}})$$ for $s\in\Bbb R.$
I know that $$ P(2^s,2^{-s}) $$ traces out the function $f(x)=\frac{1}{x}.$
- 756
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1
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Composing inverse with a function vs manipulating to have inverse in function's expression
Consider a function $f(x)=2x$ with inverse $g(x)= \frac{x}{2}$, now notice that:
$$ f(x) = 2x = 4 \frac{x}{2} = 4g$$
Or,
$$ f(x) = 4g$$
Now it looks very tempting to say:
$$ f(g) =4g \tag{1}$$
The ...
-2
votes
1
answer
240
views
What is the growth rate of $h(n)?$
$$h(n) = \#\{ \pi(x)\pi(n-x),x\le n\}$$
What is the growth rate of $h(n)?$
(the notation means find the distinct values of $h(n)$ for each $n \in \Bbb N)$
for example, plotting the point $(12,4)$ ...
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votes
2
answers
112
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How do I prove interval 𝐴⊂[0,3] exists on this integration
Let $f:[0,3] \to \mathbb{R}$ be a continuous function satisfying
$$\int_{0}^{3}x^{k}f(x)dx=0 \quad \text{for each k = 0,1,}\dots,n-1$$ and
$$\int_{0}^{3}x^{n}f(x)dx=3.$$
Then prove that there is an ...
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function plese help me [duplicate]
find $a$, if $$f(x)= \frac{x^2-ax+2}{x-2}$$ has no vertical asymptotes
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1
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if f and g are monotonically increasing functions, such that f(g(n))=O(n) and f(n)=Ω(n) then g(n)=O(n) [closed]
I have to prove this statement :
if $f$ and $g$ are monotonically increasing functions, such that $f(g(n))=O(n)$ and $f(n)=Ω(n)$ then $g(n)=O(n).$
- 11
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votes
1
answer
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proving that f is bijection from NxNxN to N [closed]
How should I construct a function to show that their exist a bijection from $\mathbb {N}$x$\mathbb {N}$x$\mathbb{N}$ to $\mathbb{N}$?
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1
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Find the relationship between $x$ and $y$ so that $y:=0\rightarrow\frac{\pi}{2}\Leftrightarrow x:=y\rightarrow\frac{\pi}{2}$
Find the relationship between $x$ and $y$ so that
$$y:=0\rightarrow\frac{\pi}{2}\Leftrightarrow x:=y\rightarrow\frac{\pi}{2}$$
I'm having trouble solving the multivariable calculus if I change the ...
user822157
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votes
1
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159
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Find all functions $f:\mathbb{R}\to\mathbb{R}$ such as $x\left[f(x+y)-f(x-y)\right]=4yf(x)$
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such as $$x\left[f(x+y)-f(x-y)\right]=4yf(x)$$
My attempt:
Let $x=0\Rightarrow 4yf(0)=0\Rightarrow f(0)=0.$
Let $y=x\Rightarrow xf(2x)=4xf(x)\Rightarrow ...
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