Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
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Infinite series, injective function and rearrangement inequality
Let $f:\mathbb{N^*}\to\mathbb{N^*}$ an injective function. Show that the following infinite series $$\sum_{i=1}^{\infty}\frac{f(n)}{n^2}$$ is divergent. I am supposed to deal with this using the ...
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What are the strategies I can use to prove $f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$?
$f^{-1}(S \cap T) = f^{-1}(S) \cap f^{-1}(T)$
I think I have to show that the LHS is a subset of the RHS and the RHS is a subset of the LHS, but I don't know how to do this exactly.
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If a function is undefined at a point, is it also discontinuous at that point?
I posted a solution here with an illustration (see below) and commented that the function was discontinuous at $x=1$, where it is undefined. Someone told me, no, it is undefined but continuous.
Now ...
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BMO1 2009/10 Q5 functional equation: $f(x)f(y) = f(x + y) + xy$
Find all functions $f$, defined on the real numbers and taking real values, which satisfy the equation $f(x)f(y) = f(x + y) + xy$ for all real numbers $x$ and $y$.
I worked out $f(0)=1$, and $f(-1)f(...
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Proof of $f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})$
I want to prove the following equation:
$$
f^{-1}(B_{1}\setminus B_{2}) = f^{-1}(B_{1})\setminus f^{-1}(B_{2})
$$
Is this a valid proof? I am not sure, because at one point I am looking at $f(x) \in ...
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Does there exist a continuous function from [0,1] to R that has uncountably many local maxima?
Does there exist a continuous function from $[0,1]$ to $R$ that has uncountably many strict local maxima?
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Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions $\sinh$, $\cosh$, and $\tanh$?
Are there parabolic and elliptical functions analogous to the circular and hyperbolic functions $\sinh, \cosh$, and $\tanh$?
Also, in matters of conic sections, are there other properties such that it ...
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Proof of properties of injective and surjective functions.
I'd like to see if these proofs are correct/have them critiqued.
Let $g: A \to B$ and $f: B \to C$ be functions. Then:
(a) If $g$ and $f$ are one-to-one, then $f \circ g$ is one-to-one.
(b) If ...
- 6,534
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Continuous unbounded but integrable functions
Many tricky exercises concern the quest for functions that satisfy particular conditions. For example, let us consider the spaces $C_p( \mathbb R), 1 \leq p < \infty$, of continuous functions on $\...
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Prove: Any open interval has the same cardinality of $\Bbb R$ (without using trigonometric functions)
I want to prove that every open interval has the same cardinality of $\Bbb R$.
The question is:
Is it enough to prove that any open interval is uncountable? If I prove it, can I say that this interval ...
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If $f(f(x))=x^2-x+1$, what is $f(0)$?
Suppose that $f\colon\mathbb{R}\to\mathbb{R}$ without any further restriction. If $f(f(x))=x^2-x+1$, how can one find $f(0)$?
Thanks in advance.
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Composition of 2 involutions
How can we prove that any bijection on any set is a composition of 2 involutions ?
Since involutions are bijections mapping elements of a set to elements of the same set, I find it weird that this ...
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Maps - question about $f(A \cup B)=f(A) \cup f(B)$ and $ f(A \cap B)=f(A) \cap f(B)$
I am struggling to prove this map statement on sets.
The statement is:
Let $f:X \rightarrow Y$ be a map.
i) $\forall_{A,B \subset X}: f(A \cup B)=f(A) \cup f(B)$
ii) $\forall_{A,B \subset X}: f(...
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Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$
I haven't been able to do this exercise:
Let $f: A \rightarrow B$ be any function. $f^{-1}(X)$ is the inverse
image of $X$. Demonstrate that if $f$ is surjective then $X = f(f^{-1}(X))$ where $X \...
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Let $f: \mathbb N \rightarrow \mathbb N$ are increasing function such that $f\left(f(n)\right)=3n$. Find $f(2017)$ [duplicate]
Let $f: \mathbb N \rightarrow \mathbb N$ are increasing function such that
$$f\left(f(n)\right)=3n$$
for any positive integer $n$.
Find $f(2017)$
My work so far:
1) If $m\not=n$ then $f(m)\...
- 17.9k
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Variable or constant, and Dependence among variables
I've been working with symbolic Mathematics for a very long time, but I still have many small questions relating to the idea of a 'constant' and the idea of change with respect to variables, and their ...
user1007028
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3
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Inverse function of $x^x$
How can I find the inverse function of $f(x) = x^x$? I cannot seem to find the inverse of this function, or any function in which there is both an $x$ in the exponent as well as the base. I have tried ...
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Graph Transformation
Knowing dealing with graph transformations come handy MANY times. I searched on google to get a comprehensive graph transformation list but couldn't find one. Some good while back I learned them all ...
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Finding the range of $f(x) = 1/((x-1)(x-2))$
I want to find the range of the following function
$$f(x) = \frac{1}{(x-1)(x-2)} $$
Is there any way to find the range of the above function? I have found one idea. But that is too critical. Please ...
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Some confusion about what a function "really is".
Despite my username, my background is mostly in functional analysis where (at least to my understanding), a function $f$ is considered as a mathematical object in its own right distinctly different ...
29
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Show that Function Compositions Are Associative
My intent is to show that a composition of bijections is also a bijection by showing the existence of an inverse. But my approach requires the associativity of function composition.
Let $f: X \...
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Proof of linear independence of $e^{at}$
Given $\left\{ a_{i}\right\} _{i=0}^{n}\subset\mathbb{R}$ which are
distinct, show that $\left\{ e^{a_{i}t}\right\} \subset C^{0}\left(\mathbb{R},\mathbb{R}\right)$,
form a linearly independent set ...
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Why aren't the graphs of $\sin(\arcsin x)$ and $\arcsin(\sin x)$ the same?
(source for above graph)
(source for above graph)
Both functions simplify to x, but why aren't the graphs the same?
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Number of non-decreasing functions?
Let $A = \{1,2,3,\dots,10\}$ and $B = \{1,2,3,\dots,20\}$.
Find the number of non-decreasing functions from $A$ to $B$.
What I tried:
Number of non-decreasing functions = (Total functions) - (Number ...
- 1,567
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Equation for a smooth staircase function
I am looking for a smooth staircase equation $f(h,w,x)$ that is a function of the step height $h$, step width $w$ in the range $x$.
I cannot use the unit step or other similar functions since they ...
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Example of a function continuous at only one point. [duplicate]
Possible Duplicate:
Find a function $f: \mathbb{R} \to \mathbb{R}$ that is continuous at precisely one point?
I want to know some example of a continuous function which is continuous at exactly ...
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Pull back image of maximal ideal under surjective ring homomorphism is maximal
Let $f :R \to S$ be a surjective ring homomorphism , $M$ be a maximal ideal of $S$ , I am writing a proof showing $f^{-1}(M)$ is a maximal ideal of $R$ , Please verify whether it is correct or not .
...
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Can the Identity Map be a repeated composition one other function?
Consider the mapping $f:x\to\frac{1}{x}, (x\ne0)$. It is trivial to see that $f(f(x))=x$.
My question is whether or not there exists a continuous map $g$ such that $g(g(g(x)))\equiv g^{3}(x)=x$? ...
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Proving that $C$ is a subset of $f^{-1}[f(C)]$
More homework help. Given the function $f:A \to B$. Let $C$ be a subset of $A$ and let $D$ be a subset of $B$.
Prove that:
$C$ is a subset of $f^{-1}[f(C)]$
So I have to show that every element ...
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Why is there no continuous square root function on $\mathbb{C}$?
I know that what taking square roots for reals, we can choose the standard square root in such a way that the square root function is continuous, with respect to the metric.
Why is that not the case ...
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