All Questions
Tagged with real-numbers functions
183
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167
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Will the roots of $p(p(x))=0$ be purely real or purely imaginary or neither?
Question
The quadratic equation $p(x)=0$ with real coefficients has purely imaginary roots. Then the equation $p(p(x))=0$ will have -
a) only purely imaginary roots
b) all real roots
c) two real and ...
CommunityBot
- 1
1
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1
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99
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If $f$ is a function such that $f(f(x))=x^{2}-1$ determine the function $f(f(f(f(x))))$
I have tried $f(f(f(f(x))))=f\left(f\left(x^{2}-1\right)\right)$ . Since we know that $f(f(x))=x^{2}-1$, we have
$$
\begin{aligned}
f\left(f\left(x^{2}-1\right)\right) &=\left(x^{2}-1\right)^{2}-1 ...
- 740
2
votes
1
answer
172
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Proving if a function is continuous and not one-one then it has many such points.
Let $g$ be a continuous function on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one; that is $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
- 50.3k
1
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0
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70
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Is this a surjection from $(0,1) \rightarrow \mathbb{R}$?
I am trying to think of creative bijections from $(0,1) \rightarrow \mathbb{R}$ that can serve as a proof of the fact that the cardinality of the interval $(0,1)$ is equal to the cardinality of $\...
- 1,186
3
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3
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211
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From which set does the number $\sqrt[3]{-1}$ belong to?
I was trying to draw the function $f(x)=\sqrt[3]{x^2(6-x)}$ by hand (I'm in my first year of engineering; having Calculus I; this drawing is actually an exercise given for my class) and used ...
1
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3
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797
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Intrepreting tuples as functions
I have been mulling over this for a while now. I am told $\mathbb R^n$ can be interpreted as a set of functions.
Take $\mathbb R^2$, for example I can see how we might interpret it as a set containing ...
- 36k
2
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2
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53
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what are all functions with $x>1$ and $y>1$ $\rho$ that follows $\rho(xy)=\frac{1}{\frac{y}{\rho(x)}+\frac{x}{\rho(y)}}$ and is continuous
What are all functions with $x>1$ and $y>1$ $\rho$ that follows $$\rho(xy)=\frac{1}{\frac{y}{\rho(x)}+\frac{x}{\rho(y)}}$$ and is continuous
If this doesn't have any solutions then prove no such ...
- 482
0
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2
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76
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Prove that function is non-monotonic and is invertible
We have $$f(x)=\begin{cases}\frac x2,&x\in\Bbb Q\\-\frac x2,&x\in\Bbb R\setminus\Bbb Q\end{cases}$$
Prove that function is non monotonic and is invertible.
I tried to take points from $\Bbb ...
- 17.7k
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3
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72
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{$\frac{1}{x}$} , $0<x\leq1$ in terms of {$x$}, $x\geq 1$, {$x$} is fractional part of $x$
$f(x)$ $=$ {$\frac{1}{x}$}, $0<x\leq1$ where {$x$} denotes the fractional part of $x$.
$g(x) =$ {$x$}, $x\geq 1$
I want an expression for $f(x)$ in terms of x and $g(x)$.
My try-
If $x\in \mathbb{Z}...
CommunityBot
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1
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47
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Prove that $\left | f(x) - a \right | \leq \frac{1}{2} \left | x - a \right |$
$f(x) = 1 - \frac{1}{x}(\sqrt{1 + x^2}-1)$
$|f'(x)| \leq \frac{1}{2}$
$a$ is a solution for $f(x) = x$ where $0.65 < a < 0,7$
The question says:
Prove that $\left | f(x) - a \right | \leq \frac{...
- 433
7
votes
2
answers
314
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What is a condition for two real functions $f,g$ to "commute", so $f(g(x))=g(f(x))$?
Say I'm given two functions $f,g$. Can I tell if they "commute" without actually trying them in the formula
$f(g(x))=g(f(x))$?
And given a function $f$, is there a way to find all functions $...
- 4,062
3
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1
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67
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How do I solve this in an understandable and direct way? [closed]
For each $i \in \Bbb N$, let $f_i: \Bbb N \mapsto \{0, 1\}$.
Let $A = \{f_i : i \in \Bbb N\}$ and $E = \{n \in \Bbb N : f_n(n) = 0\}$.
Does there exist a $f \in A$ such that $E = \{n \in\Bbb N : f(n) =...
- 341
6
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3
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159
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Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ that satisfies these conditions?
Does there exist an injection $f:\mathbb{R}\to P(\mathbb{N})$ from the reals into the power set of the naturals such that
for any $x\in\mathbb{R}$ the set $f(x)$ is infinite, and
for any distinct $x,...
- 55k
1
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2
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316
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Solving the functional equation $ xf(x) - yf(y) = (x - y)f(x + y) $ [duplicate]
I found the following functional equation:
Find all functions $f : \Bbb R \rightarrow \Bbb R $ such that:
$$ xf(x) - yf(y) = (x - y)f(x + y) \text{ for all }x, y \in \mathbb R $$
Could you please ...
- 6,766
2
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2
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381
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Is a circle a multivalued function?
I don't really understand multi-valued function. I hope one of you can make me understand it. What I've learned from google, I suppose that a multi-valued function is a binary relation that maps the ...
- 64.1k