All Questions
Tagged with real-numbers functions
183
questions
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60
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Functions problem from CGMO 2010
Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb{R}$ to $\mathbb{R}$ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $...
- 81
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1
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808
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Does $f:\mathbb{R} \rightarrow \mathbb{R}$ mean that $f(x)$ is defined for all real inputs?
$f:\mathbb{R} \rightarrow \mathbb{R}$ basically means that the domain of $f$ is the set of real numbers and the range of $f$ is the set of real numbers. However, does it mean that $f(x)$ is defined ...
- 3,184
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149
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What does trigonometric functions of real numbers really mean?
According to right triangle definition, trigonometric functions relates the angle to the ratio of sides of triangle. These functions take angles as input.
According to unit circle definition, ...
6
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2
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367
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How to strengthen $ h \big( 2 h ( x ) \big) = h ( x ) + x $ to force $ h $ to be linear?
Let $ h : \mathbb R \to \mathbb R $ be an injective function such that
$$
h \big( 2 h ( x ) \big) = h ( x ) + x
$$
for all $ x \in \mathbb R $, and $ h ( 0 ) = 0 $. What would be an as mild as ...
- 12.1k
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41
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What is the correct inverse function for $f(x) = x^2$. Question about terminology.
Is the inverse function $g_1(x)=\sqrt{x}$ or is it $g_2(x)=-\sqrt{x}$. With what terminology can we describe both of these functions?
- 2,527
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0
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17
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Looking for a sigmoid-like function with different properties
I am looking for a function that is 0 at 0, 1 at 1, increases to a predetermined $x_1$ relatively quickly, acquires a derivative close to 0 (but doesn't actually plateau) then starts increasing at $...
- 3,439
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1
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31
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How to Quantify Utility/Pleasure/Pain using the Positive Real Numbers?
I am studying about Cardinal Utility in Economics (or more generally, how to quantify pleasure and pain!)
Intuitively, I assign a positive number to pleasurable experiences, and a negative number to ...
1
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2
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72
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Prove that $0 < g(x) - \ln x \leq \ln\left(\frac{x+2}{x}\right)$ for $x > 0$
$$g(x) = \ln(x + 1 + e^{-x})$$
My question is prove that $0 < g(x) - \ln x \leq \ln\left(\frac{x+2}{x}\right)$ for $x > 0$
How do I do that?
My attempts:
I have only successfully proved the ...
- 433
1
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1
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113
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Can construct a bijection between R - Q and ( (R - Q) ∩ [0,1] )?
I've tried to show that:
$$[0,1]\sim([0,1] ∩R-Q)$$
I know from this answer :
$$[0,1]\sim R-Q$$
But how to construct a bijection between R-Q and $([0,1]∩R-Q)$ ?
I think the function would be like $f:R-...
- 382
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1
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166
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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 ...
- 175
1
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1
answer
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
$$
\begin{aligned}
f\left(f\left(x^{2}-1\right)\right) &=\left(x^{2}-1\right)^{2}-1 ...
- 55
2
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1
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169
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Proving if a function is continuous and not one-one then it has many such points.
Let $g$ be a continuous function on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one; that is $$F = \{x \in A : f(x)=f(y) \text{ for some $y \neq x$ and $y \in A$} \}$...
- 317
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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
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3
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208
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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 ...
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 ...
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