Elementary questions about functions, notation, properties, and operations such as function composition.
1
vote
0answers
17 views
Distinction between function types
Hi I'm am currently working on a question on Big O notation and to work out $O(n^k)$ of $f(n)$ you first need to know what type of function $f(n)$ is, polynomial, exponential, logarithmic. My question ...
1
vote
5answers
80 views
How do I find the image of the functions $y=2$ and $y = 2x - 6$?
The function is $y=2$, the domain is just 2? And the image of it?
I don't think I quiet understand what the image of a function means, the domain is all values that it can assume, correct?
Could you ...
2
votes
3answers
145 views
Inverse function requirements
Let f be an injective function, that is:
$f : X \rightarrow Y$
$f(a) = f(b) \implies a = b$
Now, my question is, does the following need to hold in order for function to be injective:
$(\forall x ...
0
votes
0answers
20 views
Formulating math problem with rounding / discrete step
I have this problem, that can easily be solved by simulation or numerical optimization, but I wonder how to write it as a mathematical problem? It's two pricing schemes, one cost is evaluated at ...
1
vote
1answer
39 views
Triple Integral over a disk
How do I integrate $$z = \frac{1}{x^2+y^2+1}$$ over the region above the disk $x^2+y^2 \leq R^2$?
2
votes
1answer
26 views
Show that this type of function is surjective iff it's injective.
Here's a theorem that I think intuitively makes sense, but I was hoping to prove more rigorously:
Theorem: Suppose $|A|=|B|=n$, where $n\in\mathbb{N}$. Consider the function $f:A \to B$. Then $f$ ...
4
votes
1answer
62 views
How to express $\cos(\frac{x}{k})$ and $\sin(\frac{x}{k})$ in terms of $\cos(x)$ and $\sin(x)$, respectively?
How can we express $\cos(\frac{x}{k})$ ($k \in \mathbb{N}$) in terms of $\cos(x)$?
And $\sin(\frac{x}{k})$ in terms of $\sin(x)$?
Edit
Maybe this another question helps. Is there a $T_n(x)$ ...
0
votes
1answer
33 views
Prove that $\phi: \mathbb{R}^2 \rightarrow \mathbb{R}$ is Lipschitz
I want to prove that a function $\phi: \mathbb{R}^2 \rightarrow \mathbb{R}$ is a Lipschitz function.
I proved that $|\phi(x_1,C_1) - \phi(x_2,C_1)| \leq A |x_1 - x_2|$ and $|\phi(C_2,y_1) - ...
1
vote
1answer
50 views
What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$?
Let $Y = C^1[a,b]$. What are some non-constant functions in the following sets? Are these sets a subspace of $Y = C^1[a,b]$.
a) $D = \{ y \in Y: y'(a) = 0,\, y(b) = 1\} $
b) $D = \{y \in Y : ...
13
votes
0answers
199 views
What is the algebraic structure of functions with fixed points?
So I just noticed that the set of functions with a fixed point
$$f(x_0)=x_0,$$
are closed under composition
$$(f*g)(x):=g(f(x)),$$
and with $e(x)=x$, the inverible functions even seem to form a ...
1
vote
2answers
71 views
Function that is Riemann-Stieltjes integrable but not Riemann integrable?
This is my first question, so please go easy on me :3 - I've searched, and I haven't found any questions that are particularly similar to this one.
I'm reading Rudin's Principles of Mathematical ...
2
votes
2answers
33 views
Sketching a graph under certain condtions
I got a question like this,
sketch the graph of a function that satisfies the following conditions,
the domain is [0,oo];
the range is [4,oo];
the curve passes through [0,5];
while I was ...
1
vote
1answer
55 views
Intermediate Value Property and Discontinuous Functions
This is a general question to which I need help finding a concrete example so that I may understand the concept/strategy better, and any help will be greatly appreciated.
If given a function $F$ that ...
1
vote
0answers
40 views
This is an easy question.Is this picture right?
In my text book about Fourier Analysis, $x$ ranges from $-\pi$ to $\pi$.I think when $x$ is from $-\pi$ to $0$, $\displaystyle \ln(2\sin\frac{x}{2})$ is meaningless. Why does the left part of the ...
6
votes
2answers
116 views
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
0
votes
0answers
54 views
A question on a function
Let $f: X^2 \to I=[0,1]$ be a continuous function and $M$ be any
subset of $X$. Then is the folowing claim right?
The function $f_M$ from $X$ to $I^M$ with the
compact-open topology is ...
4
votes
1answer
68 views
Square integrable function that doesn't go to zero?
I'm reading through some elementary quantum mechanics textbooks and a few authors mention that there are functions that are "there exist pathological functions that are square-integrable but do not go ...
2
votes
6answers
98 views
Function Notation
due to our national cirriculum (the way in which it was taught in high school). We just said that f(x) means a function. Though I understand this isn't necessarily correct? In high school we used ...
2
votes
3answers
406 views
How do we find the inverse of a function with $2$ variables?
$$f(m,n) = (2m+n, m+2n)$$
What do we have to do to find the inverse of this function?
I don't even know where to begin.
4
votes
3answers
50 views
$f(x)=\tanh(1+\tanh^{-1}(x))$ or $f:\tanh(x) \to \tanh(x+1)$ is a rational function?
This is (again) more a recreational/incidental question.
Playing with iteration of functions I considered the function $$ f(x) = \tanh(1+\tanh^{-1}(x)) \tag1$$ such that $$ f : \tanh(x) \to ...
0
votes
1answer
83 views
Taking limkits of tricky functions
Hi can anyone help me with this limit.
1) $\sqrt{5-\left(\frac{1}{\sqrt{1+\frac{y^2}{2}}}\right)}$ as $y\rightarrow -\infty $
I am struggling to do the first one, if it can be done using software ...
1
vote
2answers
17 views
Investigating functions - Lagranges mean value theorem
With the aid of Lagrange's formula prove the inequality :
$ \frac{a-b}{a} \leq ln \frac{a}{b} \leq \frac{a-b}{b}$
for the condition $ 0 < b \leq a$
Please guide how to proceed for this..
...
3
votes
3answers
49 views
Find no. of points where $f$ and $g$ meet.
If $f(x)=x^2$ and $g(x)=x \sin x+ \cos x$ then
(A) $f$ and $g$ agree at no points
(B) $f$ and $g$ agree at exactly one point
(C) $f$ and $g$ agree at exactly two points
(A) $f$ ...
2
votes
1answer
33 views
When in topology is $A = f^{-1} \circ f[A]$ or $B = f \circ f^{-1}[B]$ true, for an $f$ which is not one-to-one?
I'm having a bit of trouble with an example problem in the topology book I'm reading. It's problem #11 (pp 104) of the "Solved Problems" section of Chapter 7, of the Schaum's Outline for "General ...
0
votes
2answers
58 views
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}: ...
0
votes
1answer
40 views
under what conditions is f(A ∪B)=f(A) ∪f(B) and f(A∩B)=f(A)∩f(B)? [duplicate]
Does the function need to be bijective? I know for f(A∩B)=f(A)∩f(B) the function has to be injective, but what about the first equation?
17
votes
3answers
967 views
Do harmonic numbers have a “closed-form” expression?
One of the joys of high-school mathematics is summing a complicated series to get a “closed-form” expression. And of course many of us have tried summing the harmonic series $H_n =\sum \limits_{k \leq ...
0
votes
1answer
49 views
What is the property of a function called that the value is the same given an input?
I was wondering given a function like $f(x) = x +5$ that for a given $x$, the value will be the same.
1
vote
1answer
25 views
Approximating Lipschitz funtion by $C^1$ function.
Let $G:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function and $L$ its Lipschitz constant. Suppose that $G(0)=0$. It is known that $G$ is almost everywhere differentiable and $G'\in L^\infty$ with ...
0
votes
0answers
15 views
Is this function a modular function?
Is the following function a modular function?
$$\xi(x) _q=\sum_{k=1}^q \frac{1}{2q}( exp(\frac{-i2\pi (k-1)x}{q})+exp(\frac{+i2\pi (k-1)x}{q}) )$$
with $x\in\Bbb{R}$ and $q\in\Bbb{N}$ and $q\neq0$.
...
1
vote
2answers
29 views
Homeomorphism $id_M:(M,\tau_d)\rightarrow(M,\tau_h)$
I am reading thorugh some topological definitions, in my book it is stated that $id_M:(M,\tau_d)\rightarrow(M,\tau_h),x\rightarrow x$ is a Homeomorphism where
$(M,d)$ is a metric space, ...
2
votes
2answers
58 views
0
votes
1answer
25 views
percentages…
I have a sheet of plywood, say 10 sq. ft. I sell two pieces. Then, Jim bought a 5 sq ft piece while Joe bought a 2 sq. ft piece. The rest of the sheet of plywood is no good to me, so I want to ...
1
vote
6answers
86 views
A problem on range of a trigonometric function: what is the range of $\frac{\sqrt{3}\sin x}{2+\cos x}$?
What is the range of the function
$$\frac{\sqrt{3}\sin x}{2+\cos x}$$
1
vote
4answers
105 views
Is there a bijective function where $f: \mathbb Q \to \mathbb Q \setminus \{0\}$
$f(x) = {1\over x}$ should be wrong, as the function isn't defined for $0$. Another could be: $f(x) = 2^x$, but is there anything else except functions of this type?I was thinking of something with ...
0
votes
0answers
53 views
How to find the range or domain of a function?
This is a general question I'm asking, I really need it explained. Here's an example of what I mean:
The functions $f$ and $g$ are defined by
$f( x)= x^3 + 1$, $0 ≤ x ≤ 3$
$g(x)= x + 5$, ...
0
votes
2answers
35 views
Finding the derivative with functions inside, such as $g(x) = \dfrac{3x-1}{f(x)}$
With a question such as:
$$g(x) = \dfrac{3x-1}{f(x)}$$
How does one approach finding the derivative, could the Chain Rule be used?
The book, gives the answer as:
$g'(x) =\dfrac {3f(x)-(3x+1)f'(x) ...
1
vote
1answer
45 views
Proof involving functions.
Consider two functions $f\colon A \to B$ and $g\colon B \to C$. How can I prove the following?
If $f$ and $g$ are one-to-one, then the composition function $g \circ
f$ is one-to-one.
If $f$ and ...
-1
votes
0answers
24 views
Formula for the sound pressure of a pure tone [closed]
What is the formula for the sound pressure of a pure tone of 500Hz, ex- pressed as a function of time?
0
votes
1answer
77 views
Prove this proprety of $f(x)$
I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$?
Let ...
2
votes
1answer
130 views
Find power series representation for $f(x) = (4 − x)^{−3}$
Find a power series representation centered at the origin for the function $$f(x) = (4 − x)^{−3}.$$
8
votes
2answers
265 views
Prove the following property of $f(x)$?
Let $$f(x)=|a_1\sin(x)+a_2\sin(2x)+a_3\sin(3x)+...+a_n\sin(nx)|.$$
Given that $f(x)$ is less than or equal to $|\sin(x)|$ for all $x$, prove that $|a_1+a_2+a_3+....|$ is less than or equal to ...
0
votes
1answer
48 views
Additive maps modulo $1$ - what do they look like?
Recently, some convergence problems I have been considering led me to look at additive maps of the torus (which for me is the additive group $T := \mathbb{R}/\mathbb{Z}$).
A map $f:\ T \to T$ is ...
0
votes
1answer
30 views
Special type of convexity
Does anyone know if there is a name for a function $f:W\rightarrow\mathbb{R}$ ($W$ is a vector space) that satisfies
$$
f\left(\theta x + \left(1-\theta\right) y\right) \leq \theta^\alpha ...
1
vote
2answers
77 views
Show that equation has no solution in $(0,2\pi)$
Hi I want to show that the equation $2=2 \cos(x)+x \sin(x) $ has no solution in $(0,2 \pi)$. Since it is algebraically impossible to solve this equation for $x$ I wanted to ask you whether one of you ...
6
votes
4answers
291 views
$f: \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2)=5$
The function $f : \mathbb{R} \to \mathbb{R}$ satisfies $(x-2)f(x)-(x+1)f(x-1) = 3$. Evaluate $f(2013)$, given that $f(2) = 5$.
0
votes
2answers
40 views
Show uniform convergence of indefinite function series
How can i show uniform convergence on function series like this one: $\sum\limits_{k=1}^{\infty} (\sqrt{1-x^{n}}-1)$ ? I have a given interval of [0 / 0.5]
I thought about using the Weierstrass ...
1
vote
2answers
39 views
Find a polar representation for a curve.
I have the following curve:
$(x^2 + y^2)^2 - 4x(x^2 + y^2) = 4y^2$ and I have to find its polar representation.
I don't know how. I'd like to get help .. thanks in advance.
1
vote
1answer
38 views
composite function with conditional IF
I've been wrapping my head around my Computer and Logic Essentials class, I can do most composite functions, however there is one question that I'm confused with.
It has an if statement inside it:
...
2
votes
2answers
130 views
Is there any way to find minimum without the use of derivatve?
The function is: $$\sqrt{(x+1)^2+\left(2x^2-\frac{1}{4}\right)^2}$$
It simplifies to: $$\sqrt{4x^4+2x+\frac{17}{16}}$$
