All Questions
Tagged with solution-verification functions
1,071 questions
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About inflection points and change of sign
The problem states: suppose you have a function $f(x)$ and a line $g(x)$, $f(x)$ intersects $g(x)$ 3 times in: $x_1<x_2<x_3$. $f(x)$ also has 3 derivatives and its third doesn't vanish in $[x_1,...
- 723
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2
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If $\lfloor 2x\rfloor -2\lfloor x\rfloor=\lambda$ then $\lambda=?$
Question:
$\lfloor2x\rfloor-2\lfloor x\rfloor=\lambda$ where $\lfloor .\rfloor$ represents floor function and {.} represents fractional part of a real number then
A) $\lambda=1\;\forall \;x\in\mathbb ...
- 10.6k
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Find the value of $\frac{d}{dx}f^{-1} (x)\vert_{x=1}$ if $f^{-1}(x)$ cannot formed explicitly.
Find the value $\left.\dfrac{d}{dx}f^{-1}(x)\right\rvert_{x=1}$ if $f(x)=x^3+e^x$.
\begin{align}
&f(x)=x^3+e^x\\
\iff& x=f^{-1}(x^3+e^x)\\
\iff& 1=(3x^2+e^x)\dfrac{d}{dx}f^{-1}(x^3+e^x)\\
\...
- 2,696
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Verify uncountability of the set of sequences of natural numbers whose $n$th value is a multiple of all values at indices which divide $n$
A countability exercise consists in proving whether the set $A$ is countable or uncountable, where $A$ is the set of sequences of natural numbers whose $n$th value is a multiple of all values at ...
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Properties of a function from a finite to a finite set as a way to determine cardinality
In my discrete math course, my professor loosely mentioned 2 following theorems
For any finite sets A, B, a function $f:A\to B$ is defined. If f is injective then $|A|\le|B|$, and if f is surjective, ...
- 146
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Tao's Analysis I, Exercise 3.3.1
Definition 3.3.7 (Equality of functions): Two function $f : X \rightarrow Y$, $g : X \rightarrow Y$ with the same domain and range are said to be equal, $f = g$, if and only if $f(x) = g(x)$ for all $...
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Is $f(x)>0$ for $x>0?$ [closed]
The following question is taken from the practice set of JEE.
Question:
$f(x)$ is defined for $x>-1$ and has a continuous derivative. $f$ satisfies $f(0)=1, f'(0)=0, (1+f(x))f''(x)=1+x$. If $x$ is ...
- 10.6k
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Error in my understanding of $\int^{s}_{1}f(x)dx \geq \sum^{s}_{k = 2}f(k)$ from Hassani's 'Mathematical Physics'?
Convince yourself that $\int^{s}_{1}f(x)dx \geq \sum^{s}_{k = 2}f(k)$ for any continuous function $f(x)$, and apply it to part (c) to get...
The statement above is from Section 1.6, Problem 1.7 d) of ...
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Question about continuity of inverses
I got this question from a friend and I think it's a bit of weird question but here it is. It's not complicated but I'm not sure how to think about inverses of functions in these non-standard ...
user1148590
2
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1
answer
41
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What is correct and what is wrong in the derivative of the composition of two functions?
Let :$f(x,y)=f(u,v)$
In the lecture at my university :
$$\frac{\partial^2 f}{\partial x^2}=\frac{\partial^2 f}{\partial u^2}\left({\frac{\partial u}{\partial x}}\right)^2+2\frac{\partial^2 f}{\partial ...
- 3,983
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Find fundamental period of $\sin(5π× g(3x))$ and $\sin^2(5π × g(3x))$. $g(x)$ denotes fractional part of $x$. (Verifying my method)
The question- Find fundamental period of $\sin(5π× g(3x))$ and $\sin^2(5π × g(3x))$.
$g(x)$ denotes fractional part of $x$
Note- (fractional part of x = $x-[x]$), $[x]$ is greatest integer function)
...
- 372
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Proof of equivalent definitions of injectivity
Define a function $f: X\rightarrow Y$
Then the following statements are equivalent:
i)$f$ is injective.
ii)$f^{-1}(f(A))=A,A \subset X$
iii) $f(A \cap B)=f(A)\cap f(B)$
Proof:
(I have edited this ...
- 403
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Properties of Image of the Pre-Image of the Image of a Subset of the Domain & Pre-Image of the Image of the Pre-Image of a Subset of the Codomain [duplicate]
I know the title somehow sounds like a tongue twister but, mathematically it is much easier to explain. The definition of image and inverse image of a subset is as provided in the textbook on section ...
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Questions about when to prove for the special case of the empty set
So the lecture notes I am reading proved a theorem: A function $f \colon A \to B$, with $A \neq \emptyset$ is injective if and only if it has a left inverse, and it's surjective if and only if it has ...
- 1,513
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f refers to the function with domain $[0, 2]$ and range $[0, 1]$. Sketch the graphs of $y=2f(x)$ and $y=f(2x)$.
The problem:
f refers to the function with domain $[0, 2]$ and range $[0, 1]$ whose graph is shown in the figure. Sketch the graphs of the indicated functions, and specify their domains and ranges.
$...
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