Elementary questions about functions, notation, properties, and operations such as function composition.
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 ...
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 ...
4
votes
1answer
47 views
Suppose that the function f(x)
Suppose that a function $f(x)$ defined on $[0,1]$ satisfies
$f(1/n)\to 0$ as $n\to\infty$. Can we say that $f(x)\to 0$ as $x\to 0^+$ if $f$ is continuous on $[0,1]$ ? and again
is it true $f(x)\to ...
-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
0answers
52 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
$f( x)= x^3 + 1$, $0 ≤ x ≤ 3$
$g(x)= x + 5$, ...
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
$$\frac{\sqrt{3}\sin x}{2+\cos x}$$
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 ...
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 ...
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
$$
f\left(\theta x + \left(1-\theta\right) y\right) \leq \theta^\alpha ...
6
votes
2answers
116 views
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Find all function $f:\mathbb{R}\mapsto\mathbb{R}$ such that $f(x^2+y^2)=f(x+y)f(x-y)$.
Some solutions I found are $f\equiv0,f\equiv1$, $f(x)=0$ if $x\neq0$ and $f(x)=1$ if $x=0$.
0
votes
1answer
77 views
Prove this proprety of $f(x)$
I've asked this question before a long time ago, but I didn't get a complete answer. This is the link to the incomplete answer: Prove the following property of $f(x)$?
Let ...
0
votes
2answers
40 views
Show uniform convergence of indefinite function series
How can i show uniform convergence on function series like this one: $\sum\limits_{k=1}^{\infty} (\sqrt{1-x^{n}}-1)$ ? I have a given interval of [0 / 0.5]
I thought about using the Weierstrass ...
1
vote
2answers
39 views
Find a polar representation for a curve.
I have the following curve:
$(x^2 + y^2)^2 - 4x(x^2 + y^2) = 4y^2$ and I have to find its polar representation.
I don't know how. I'd like to get help .. thanks in advance.
1
vote
1answer
36 views
composite function with conditional IF
I've been wrapping my head around my Computer and Logic Essentials class, I can do most composite functions, however there is one question that I'm confused with.
It has an if statement inside it:
...
0
votes
1answer
35 views
Question about basic properties of linear functions
It is proveable that if
$$f(x)=\frac{f(x-k)+f(x+k)}{2}$$
and $f(0)=0 $ for all $x,k\in \mathbb{R} $
Then,$f(x)$ is linear
But my question is, if it is true for all $x \in \mathbb{R}$ but only for ...
2
votes
2answers
130 views
Is there any way to find minimum without the use of derivatve?
The function is: $$\sqrt{(x+1)^2+\left(2x^2-\frac{1}{4}\right)^2}$$
It simplifies to: $$\sqrt{4x^4+2x+\frac{17}{16}}$$
2
votes
1answer
98 views
Functional equation
Can someone help me please with this problem?
If the function
$f:\mathbb{R}^+\rightarrow\mathbb{R}$ satisfies the equation
$f\Big(\frac{x+y}{2}\Big)+f\Big(\frac{2xy}{x+y}\Big)=
f(x)+f(y)$,
then it ...
5
votes
2answers
64 views
Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ . . .
Find all functions $f$ that assign a real number $f(x)$ to every real number $x$ such that
$$(x+y)f(x)+f(y^2)=(x+y)f(y)+f(x^2)$$
I've tried subbing in heaps of values but I keep getting things like ...
1
vote
0answers
28 views
What's the most straight forward way to show that a function is increasing?
I am trying to show that:
$$\frac{2}{n}\log\Gamma(\frac{x}{2}) - \log\Gamma(\frac{x+n-1}{n})$$
is an increasing function for $x \ge 5$ and $n > 2$
One way to do this would be to show that ...
2
votes
5answers
162 views
Looking for a function with $f(0) = 0$, $f'(0) = 1$ and $\lim\limits_{x\to\infty}f(x)=1$
I'm looking for a monotonic, continuous function function with these properties:
$$f(0) = 0$$
$$f'(0) = 1$$
$$\lim\limits_{x\to\infty}f(x)=1$$
I also need a coefficient that allows me to configure ...
8
votes
1answer
159 views
Function mapping challange
For a given set $A=\{1, 2, 3, 4, \ldots, n\}$, find the number of non-constant
mappings (associations ) $f$ from $A$ to $A$ such that $f(k) \leq f(k + 1)$
and $f(k) = f(f(k + 1))$.
This is ...
0
votes
1answer
52 views
basic mathematical function
I have a basic question.
When I store 10000 objects in a locker for 10000000 years, I lose 1 object. How many objects should I have, if I want to lose the same 1 object in 1 year. Is it ...
0
votes
2answers
63 views
How to find when function is increasing?
When is the function defined by $f(x)=x^2+e^{-2x}$ increasing? I know you have to take the derivative and use certain values of $x$, but I am confused on how to do this particular problem, and I ...
2
votes
1answer
86 views
Functional equation $f(y/x)=xf(y)-yf(x)$
Are there any solutions to $f:\mathbb{R} \backslash \{0\} \rightarrow \mathbb{R}$ with $f(\frac{y}{x})=x \cdot f(y)-y \cdot f(x)$ other than $f(x)=0$ for all $x \in \mathbb{R}$?
I already have found ...
1
vote
3answers
46 views
Composition of Continous function is continuous
I have an example, prove that the function y = |cosx| is continuous.
We can make two function viz. let g(x) = |x| f(x) = cosx
As we know that |x| is continuous function and cosx is also continuous ...
0
votes
1answer
62 views
Using Rolle's theorem to deduce a property of repeated roots
I've been trying for a while now to figure out how to use Rolle's theorem to show that if $x_1$ is a repeated root of the equation $f(x)=0$, then $x_1$ is also a root of the equation $f '(x)=0$. Now, ...
1
vote
1answer
61 views
Exponential Approximation of the Modified Bessel Function of first kind (Equation)
Hello Dear Sir/Madam,
Could someone please tell me the reference or equation for "Exponential Approximation of the Modified Bessel Function of first kind" or any matlab function to compute it. My ...
1
vote
1answer
26 views
Visual difference between strictly concave and not strictly concave
Ok, this is an elementary question that has been bothering me for a while. I have a function like
$z(x,y)=x^{0.6}y^{0.4}$.
Defined for $x,y\in R^{++}$
Plainly spoken, this function is concave but ...
6
votes
6answers
183 views
Representing the function $\mathbb Z_9\to\mathbb Z_9$, $f(0) = 1$, $f(1) = \ldots = f(8) = 0$ as a polynomial in $\mathbb Z_9[x]$
Let $\mathbb Z_9=\left\{0,1,2,3,4,5,6,7,8\right\}$ be the set of integers modulo 9 and $f:\mathbb Z_9 \rightarrow \mathbb Z_9$ be a function.
Assume $f(0)=1$, $f(1)=f(2)=...=f(8)=0$. What is the ...
3
votes
0answers
60 views
How do zeros on the complex plane affect the real number line?
Let's say there is a real-valued "signal" that you can only measure at discrete points $f(x)$. You have a theory that this signal is the result of an analytic function $f(z)$ on the complex plane but, ...
0
votes
2answers
40 views
Finding the points of disctontinuity of greatest integer function
Let $f(x) = [x^3-3]$ where x is the greatest integer function. Then find the number of points in the interval (1,2) where this function is discontinuous ?
Please suggest , how to proceed in this ...
0
votes
2answers
32 views
Determine the area of the region bounded by $y=2e^x$, $ y=e^{2x}$ and $x=0$
$$y_1 = 2e^x$$
$$y_2 = e^{2x}$$
$$x=0$$
I was thinking of finding the $x$-intercepts first, so $2e^x= e^{2x}$.
What is next?
1
vote
2answers
26 views
Simple Filter Function
I'm trying to find a function $f(x)$ when $x>1$ which gives $0$ when $x=1$ and $200$ for all positive integers $x>1$.
i.e.
$f(1) = 0 $
$f(2) = 200$
$f(3) = 200 $
$f(4) = 200$
and etc..
...
1
vote
1answer
57 views
nth term test for divergence - help
$$\sum_{n=1}^{\infty} \left(\dfrac{n}{n+1}\right)^n$$
to show that this diverges should I use the $n^{th}$ term test?
So far I have substituted infinity for $n$. Could I use L'hopital's rule to ...
1
vote
1answer
80 views
Use the Mean Value Theorem to prove the Binomial Inequality?
Binomial Inequality: $\forall x \in \mathbb{R}, x\geq-1, \forall n \in \mathbb{N}: (1+x)^{n} \geq 1+nx$
I need to prove this using the Mean Value Theorem: If $f$ is continuous on $[a,b]$ and ...
0
votes
1answer
45 views
Supremum and infimum for this set - help needed
what is the set S bounded by? how do I do these questions? can someone please show me an exemplar solution that I could follow - thank you
0
votes
0answers
45 views
mean value theorem question!
Let a function $f$ be a contraction function in $[-a,a]$. Derive conditions on $f$ such that $|f(x)|\leq|x|$ for all $x\in[-a,a]$.
I need to answer this question using the mean value theorem but ...
1
vote
2answers
35 views
Initial Value Problems?
Suppose $f:\mathbb{R}\rightarrow\mathbb{R}$ is a solution of the initial value problem $f'=f; f(0)=1$I need to answer the following questions:(i)Fix $y\in\mathbb{R}$ and set ...
0
votes
0answers
59 views
plot $(x,y)$ coordinates to a sphere
I have a planar mesh of hexagons (or any other shape) that I want to bend into the shape of a half-sphere. For this purpose I want to loop through each vertice in my mesh and find the proper z ...
0
votes
1answer
37 views
A smooth or analytic function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
In A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer, I asked about continuous function. This time, I would like to ask the ...
0
votes
1answer
71 views
For which values of $a$ is $f(x) = \cos(ax)$ a contraction mapping?
I am studying contraction mapping and got stuck on this question:
Consider the following function:
$$f(x)=\cos(ax),\ a,x \in \mathbb{R}.$$
Find values of $a$ for which $f$ is a ...
0
votes
2answers
44 views
A continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ where $n$ is prime and $k$ is integer
Can we find a continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has fixed points at $kn^2$ for every integer $k$ and every prime number $n$? And other points other than these are not fixed ...
1
vote
1answer
34 views
Definition of correspondence
A one-to-one correspondence is an alternative name for a bijection between two sets, but to what does the term 'correspondence' alone refer? As far as I can see, it seems to be another term for ...
0
votes
3answers
47 views
Domain or range?
I am doing some questions on contraction mapping and the first part of the question is:
I am not sure what the question is asking for and how to do it.
Any help is hugely appreciated! thank you
3
votes
2answers
49 views
Can it be shown that a set X is infinite if and only if there exists some $F:X\to X$ that is an injection but not a surjection?
If the function is not surjective, then at least one element of the codomain has no pre-image. However, because F is a function, every element in the domain is mapped to something in the codomain. ...
0
votes
3answers
39 views
Can there be a non-polynomial continuous function from $\mathbb{R}$ to $\mathbb{R}$ that has multiple zero-valued points?
1) Is there any non-polynomial function from $\mathbb{R}$ to $\mathbb{R}$ that has more than one zero-valued point in domain?
2) Is there any non-polynomial function from $\mathbb{R}$ to $\mathbb{R}$ ...
11
votes
2answers
349 views
Does $f(x) = f(2x)$ for all real $x$, imply that $f(x)$ is a constant function?
If a Continuous function $f(x)$ satisfies $f(x) = f(2x)$, for all real $x$, then does $f(x)$ necessarily have to be constant function? If so, how do you prove it? If not any counter examples?
1
vote
2answers
37 views
Looking for function with specific properties
I need a function $f$ that is arbitrarly times differentiable and which has integral
$$\int _a^b f(x) dx $$ strictly positive (where $a$ and $b$ are fixed), and for all derivatives, we have ...
2
votes
0answers
14 views
Repertoire method for solving recursions
I am trying to solve this four parameter recurrence from exercise 1.16 in Concrete Mathematics:
$g(1) = \alpha;$
$g(2n+j) = 3g(n) + γn + β_j$ : j = 0,1 and n >= 1
I have assumed the closed form to ...
