Elementary questions about functions, notation, properties, and operations such as function composition.

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2answers
21 views

Clarification of the notation $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$

I have a question that uses the following notation: the function $f: \mathbb{R} \setminus\{3\} \to\mathbb{R}\setminus\{2\}$ is defined by $$f(x)=\frac{2x-3}{x-3}.$$ I understand that the left side ...
0
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2answers
8 views

Show that f has another inflection point and compute the (x, y)-coordinates of this other point of inflection?

Consider the function $f(x) = ax^4 − 8x^3 + b$. Assume that $(x, y) = (2, 8)$ is an inflection point of this function. Show that f has another inflection point and compute the $(x, y)$-coordinates of ...
1
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0answers
14 views

Need to find approx. formula for sequence data

I have an IT problem to solve, but I came into a blind spot when it came to mathematics.. Ill give a compact background, maybe it helps. The sequence, that I'll have to reverse, is the size of the ...
0
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4answers
27 views

Injective? Surjective? Bijective? None?

Is the following diagram representative of an injective, surjective, or bijective function? (or none) The reason why I'm asking is because by the definitions of injectivity and surjectivity, this ...
0
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1answer
21 views

number of surjective functions

Let $A, B$ sets, $|A| = n, |B| = r,~ 1 \le r \le n$. Prove that there are $\displaystyle \sum_{k_1 + \cdots + k_r =~n \atop k_1,\ldots,k_r \in \mathbb N} \frac{n!}{k_1! \cdots k_r!}$ surjective ...
1
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1answer
37 views

If $f(x-1) +f(x+3) = f(x+1)+ f(x+5)$, find the period of $f(x)$.

I have an assignment full of questions like these, and I know that using some suitable substitution like replacing $x$ with $(x-1)$ and simplyifying it, I will be able to arrive at a stage where the ...
2
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4answers
91 views

Solving $\;2^{\large \cos x} = \sin x$

$$2^{\large \cos x} = |\sin x|$$ Solve the equation. I found just one solution $\cos x= 0$ and are there any other solutions. Right hand side is modulus $\sin x$.
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0answers
35 views

Show $f:\mathbb{N}\times\mathbb{N}\to\mathbb{N}\times\mathbb{N}$, defined by $f(x,y)=(x+y,2x-3y)$ is one-to-one [on hold]

I need help with this function. Show $f:\mathbb{N}\to\mathbb{N}$, defined by $f(x,y)=(x+y,2x-3y)$ is one-to-one I dont understand this.
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0answers
34 views

The Riemann Zeta Function Works [on hold]

1 - Any counterexamples known for the Riemann Zeta Function? 2 - How to generalize the following? Here we have the visualization of the Riemann Zeta Function 3D Plot and the plane. We can observe ...
0
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1answer
34 views

Inequality depending on a series and a function

I know that $ \forall n\in N, a_0 + a_1 + ... + a_n = 1$, with $a_0, a_1, ... a_n > 0$ and $f(t) = a_0t^n+a_1t^{n-1}+...+a_n, \forall t\in R$ I have to prove that $$ f\left(\sqrt{x^3} \right)* ...
1
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0answers
23 views

function 3 digit arithmetic

I want ask this question correct? If wrong, please show the way to calculate. I am some confusion the 3 digit arithmetic, can explain it to me?
0
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1answer
25 views

How to tell if the following function is one to one

Let f:A→B where A = X∪Y with X∩Y=∅. If f|x and f|y are one-to-one, does it follow that f is one-to-one? I am unsure how to figure this out. I have gathered from the info provided that X and Y are ...
0
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1answer
8 views

How long do I take to do each piece of work if I have to clear one claim batch every 30 mins.

One day this week I worked 7hrs 5 minutes. I have to prepare batches of claims ready for scanning. we have a target to do of 1 batch every 30 mins. I normally clear 21 batches a day so how long ...
0
votes
1answer
36 views

Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$…

Problem : Find the range of function $f(x) =\cos(\sin(\ln(\frac{x^2+e}{x^2+1})))+\sin(\cos(\ln(\frac{x^2+e}{x^2+1})))$ My approach : maximum value of the function is when denominator term is ...
0
votes
1answer
16 views

Prove concavity without testing the second derivative..

Consider a function $F(L)=(L-L^2a)T^{L-1}$, where $0<L<\frac{1}{a}$. The constants $a$ and $T$ may take values over $]0,1[$ and $[0.01,0.1]$, respectively. The first derivative of $F$: ...
1
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1answer
12 views

Domain of dericative of a function [on hold]

Give an example of a function the domain of whose derivative is a PROPER subset of its own domain.
2
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6answers
78 views

In f:A→B, surely A and B are redundant

If a function is a set of ordered pairs, it defines its own proper domain and codomain. f determines A; a minimal B. If we extend B, do we have a different function?
0
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1answer
22 views

Alternative Definition for Injective Function?

I came up with an alternative definition of an injective function and would like to know if it's correct and how to prove it if it is, or why it's not correct if it isn't. f:A→B is injective if ...
1
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2answers
69 views

Why is $x^5 \sin x$ an odd function?

Why is $x^5 \sin x$ an odd function? Is the result just wrong? Because $f(-x)= (-x)(-x)(-x) \sin(-x) = (-x)(-x)(-x)(-x)(-x) (-\sin x) = (-x^5)(-\sin x) = x^5 \sin x$
1
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2answers
40 views

Showing one-one onto

I wanted to find the values of $a$ for which the function $f:\mathbb R\to \mathbb R$ defined by $f(x)=ax+\sin x$ is bijective. Any hint will be appreciated.
1
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1answer
19 views

inclusion of sets and inverse function

Is it true that for any function $f$, and sets $S_1,S_2$ such that $f:S1\rightarrow S2$, if g is the inverse of f $g = f^{-1}$, then $f(g[S1])\subseteq S1\subseteq g(f[S1])$?. If yes, is there a ...
1
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2answers
19 views

Range of a composite function

How to find the range of functions like $f(x)=\sin (x) ^{sin(x)}$ on $(0,\Pi)$? Usually, I find the inverse and then find the domain of the inverse function for the range of the original function, ...
1
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1answer
27 views

Transpose of composition of functions

I am trying to find an alternative proof that $(AB)^t=B^tA^t$. I think it is possible to do by showing that: $(g \circ f)^t=f^t \circ g^t$ where f,g are linear maps between vector spaces. I am ...
0
votes
0answers
17 views

Lagging or Leading trigonometric functions.

Consider the function $f(x) = 2 * \sin(0.5 * x)$. Now suppose I want to create a function which is similar to the mentioned but to "lag" the mentioned function by $45$ degree angle, then which of the ...
2
votes
7answers
104 views

$f\left(x + \frac1x\right)= x^3+x^{-3},$ find $f(x)$

$$f\left(x + \frac1x\right)= x^3+x^{-3},$$ find $f(x)$. What i do know at this state is that.. express x as a function of y : $y= x + 1/x$ $x^2−xy+1=0$ Quad formula: $x= (y ± \sqrt {y^2-4}) / 2$ ...
0
votes
4answers
52 views

Proving an equation is a fuction

Prove that the equation $y^3 + 3xy -5x^3 + 1 = 0$ defines $y$ as a function of $x$ for all $x$ in the real numbers.
0
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0answers
10 views

Eigenvalues and Positive-Definiteness of the Hessian Matrix

Suppose we have a function $f \in C^{2}$ and the Hessian defined as follows: $Hf(x,y)(h) = \displaystyle\frac{1}{2} \begin{pmatrix} h_{1} & h_{2} \\ \end{pmatrix} \begin{pmatrix} f_{xx} ...
0
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2answers
29 views

How to prove functions are odd and even

Show that any function f on [-a,a] where a is a positive constant, can be written as the sum of an even and an odd function?
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0answers
17 views

Unbounded Function

I am trying to find vales of $a >0$ s.t the function is unbounded on $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
0
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2answers
25 views

Notational difference, functions and mappings, talking about sets and classes

A Function is a set of pairs such that no two pairs have the same first member. My question summarized: What if I want to consider proper classes of pairs? The closest question to mine I could ...
7
votes
2answers
59 views

Function composition: $f^{653}(56)=?$

Let $f(x) = \frac1{(1-x)}$. Define the function $f^r$ to be $f^r(x) = f(f(f(...f(f(x)))))$. Find $f^{653}(56)$. What I've done: I started with r=1,2,3 and noticed the following pattern: $$f^r(x)= ...
0
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1answer
13 views

Linearization of a function in the point 0, 0

The linearization of the function $ f(x, y) = 1 + 2(x + 1) + 3(y + 1) + 4x^2 + 5y^2 $ in the point (0, 0) is given by: $ L(x, y) = 6 + 2x + 3y $ I know this is true, but how does one come to this ...
1
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3answers
21 views

For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges?

For real valued function $f$ define $$S(f)=\{x:x>0,f(x)=x\}$$ For which of the following functions the $\sum_{x\in S(f)}\frac{1}{x}$ converges? $\tan x,\tan^2x,\tan{\sqrt{x}},\sqrt{\tan x},\tan ...
1
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1answer
23 views

Composite Functions

$f(x)= \dfrac{1}{10x+17}+13$ $g(x)= \dfrac{1}{9x-6}$ I need to find $f(g(x)).$ How do I do this? I keep on getting it wrong. The correct answer is $\dfrac{1998x-1202}{153x-92}$. But I am unsure how ...
0
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1answer
14 views

A question regarding real valued function

I have a question regarding real-valued function: Which of the following cannot possibly be the rule of any real-valued function? A) $y=\sqrt{x-1}$ B) $y=\sqrt{x-1}+\sqrt[3]{2+x}$ C) ...
1
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1answer
11 views

Finding inverses of two functions and their compositions to solve for unknown.

$$f(x) = 23x + 27,\;\; g(x) = 12x - d$$ I've found $f^{-1}(x),$ and $\,g^{-1}(x)$, but I don't know how to solve for $d$, given $$f^{-1}(g^{-1}(x)) = g^{-1}(f^{-1}(x)).$$ How do I do this please?
1
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2answers
20 views

Which of the following is constant?

If $f,g$ are continuous real valued functions such that $f\circ g$ is constant then which of the following must be constant? $$f,g,g\circ f$$ I think when $f\circ g$ is constant then at least one of ...
0
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0answers
11 views

composition of Riemann integrable functions.

I have two functions: f:[a,b]->R and g:[c,d]->R where a My question is if it follows that g o f (the composite function of f,g) is Riemann integrable as well?
1
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3answers
52 views

Problem about a bijective map from $\mathbb R^2 \rightarrow (0,1)$ [on hold]

Does there exist a bijective map from $\mathbb R^2 \rightarrow (0,1)$? What will be the mapping?
1
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1answer
43 views

Prove whether a particular function is concave

Given the following equation: $$V(w) = - \frac{\alpha}{2} \left[ y_1(w) + y_2(w) + \int _{-\infty}^{+\infty} \vert y_1(w) - y_2(w) - x\vert f_{T1}(x)dx\right] \\- \beta \int _{w - y_1(w)} ^{+\infty} ...
2
votes
3answers
187 views

Homeomorphism(topological spaces) version of Cantor–Bernstein–Schroeder theorem

Let $A$ , $B$ be topological spaces such that there for some subset $D$ of $B$ there is a homeomorphism form $A$ to $D$ and for some subset $E$ of $A$ there is a homeomorphism form $B$ to $E$ ; then ...
0
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0answers
6 views

MR Function Question

I have a question about MR Function below: P = 1/Q^2 + 3Q + 1 Find the MR Function and Evaluate it at Q = 4 Are you guys able to elaborate? I've never done these types of questions before, am i ...
0
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1answer
31 views

Why have we made a function to be many to one and not one to many? [on hold]

We have allowed function to only relate many to one but not one to many. Why haven't we included sin(x) to be a function? Is it just for simplicity? Also, I've seen someone quote a function not even ...
2
votes
0answers
21 views

Real Analysis Differential functions

I am currently working through an exercise set and I am a little stuck on the following question: For $a > 0$, define a function $f_a(x)= \begin{cases} x^a \sin(1/x), &x \ne 0\\0, &x=0 ...
3
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1answer
78 views

How to find the function $f$ that satisfies $f(x, y) = f(x^{-1}, y^{-1})^{-1}$ and $f(x, y)$ is $\approx$ $average(x, y)$?

Fist of all, I'm a programmer, not a mathematician, and I'm sorry for my non native English. And I'm sorry if the question is not appropriate, it is my first time here. Or if the question has no ...
1
vote
1answer
41 views

Lyapunov exponent for simple functions

Context: We know that $\cos(x)$ if taken recursively on itself, converges to the Dottie number, which is the function's stable fixed point then. On the other hand, for a function like $f(x)=3x$, ...
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0answers
11 views

“For each of the three surfaces, what is the stopping distance for a car traveling at 80km/h?” [on hold]

"The stopping distance of a car on dry asphalt can be modeled by the function $d(s) = 0.006s^2$, where $d(s)$ is the stopping distance, in meters, and $s$ is the speed of the car in kilometers per ...
1
vote
1answer
18 views

Drawing toral automorphisms

How do we set about drawing a toral automorphism as in figure 5.1 in the picture above. How do we know where the points highlighted in yellow are? What happens if the eigenvalue (Im guessing some ...
0
votes
3answers
26 views

Limit of a function w

If $f(x, y)$ is a continus function, defined in whole $\mathbb R^2$, then the limit $$\lim_{(x, y)\rightarrow(2,2)}f(x, y)(x-1)(y-2) $$ The solution is $0$, but how? A very elaborative explanation ...
4
votes
2answers
37 views

Making a piecewise function continuous and differentiable at point

Problem: Let $f(x) = \left\{ \begin{array}{lr} \frac{\arctan(x)}{(1+x)^2} & : x \geq 0\\ Ae^x + B & : x < 0 \end{array} \right. $ Find $A$ and $B$ such that the function is ...