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

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12 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
58 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 $f(-x)= (-x)(-x)(-x) \sin(-x) = (-x)(-x)(-x)(-x)(-x) (-\sin x) = (-x^5)(-\sin x) = x^5 \sin x$
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2answers
36 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.
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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 ...
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2answers
18 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, ...
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1answer
25 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 ...
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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 ...
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6answers
79 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: $x= (y ± \sqrt {y^2-4}) / 2$ ...
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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.
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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} ...
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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 $[0,1]$ $f_a(x)= \begin{cases} x^{a-2}(ax\sin(1/x)-\cos(1/x)), & x\neq 0 \\ 0, & x =0 \end{cases}$
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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 ...
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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)= ...
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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 ...
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3answers
19 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 ...
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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) ...
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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?
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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 ...
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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?
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3answers
48 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?
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1answer
39 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} ...
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3answers
186 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 ...
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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
30 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
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0answers
20 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 ...
2
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1answer
60 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 ...
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1answer
38 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 ...
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1answer
17 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 ...
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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
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2answers
30 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 ...
4
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6answers
66 views

Domain of $\frac{1}{\frac{1}{x}}$

Let $f(x)=\frac{1}{x}$, then we have $f^{-1}(x)=\frac{1}{x}$. So $f(f^{-1})=\frac{1}{\frac{1}{x}}=x$. My question is, what is the domain of $f(f^{-1})(x)$? is it everything? or everything but zero? ...
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3answers
18 views

Intersection of inverse images

Given $A$ and $B$ is the subset of $C$ and $f:C\mapsto D$, $$f(A\cap B)\subseteq f(A) \cap f(B)$$ and the equality holds if the function is injective. But why for the inverse, suppose that $E$ and ...
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1answer
15 views

Integral of a normal function multiplied by heaviside and delta functions

$\int_{-\infty}^{\infty} e^{2t}u(\tau - t)t^{2}\delta(t)dt$ Hi! How would I go about computing this integral? I understand I can change one of the integration limits and eliminate the heaviside ...
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0answers
15 views

Questions about functions, their domains and codomains.

I am playing around with equations about functions in general and have some questions. Question 1 If I have some functions $f,g\colon X^2 \rightarrow Y$ such that $f(x,y) = g(x,c)/g(y,c)$ then can ...
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1answer
12 views

Composing Piecewise Functions

I was wondering how to compose piecewise functions. On a practice exam I was reading, a question asks what F(F(x)) will look like if F(x)= 2x if x<1/2 and = 2-2x if x>=1/2. Would I just ...
0
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1answer
39 views

How to calculate “general” integral $\int\limits_{a}^{b}f(x)^2dx$?

How to calculate "general" integral: $\int\limits_{a}^{b}f(x)^2dx$?
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1answer
37 views

What is this integration “method” name?

I see that people often write this equality: $$\int\limits_a^bf(x)\,\mathrm dx=\int\limits_{f(a)}^{f(b)}f(x)\,\mathrm df(x)$$ when dealing with functins in general, that is when something is trying ...
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1answer
25 views

Arccos function [on hold]

How to look for condition (limits of x): Or, how to get domain limits for $-1 < = \frac{( x-1)} {(2 x +7) } < = 1 $ for using in -1 <= argument <= 1 ( to get real values of ...
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1answer
24 views

Inequality from a property of convexity

Let $f:\mathbb{R}\to\mathbb{R}$ be a convex function. How can I prove that for each $x$, there is $c$ such that $f(x)+c(y-x)\leq f(y)$ for all $y$? One of the difficulties to solve is $f$ does not ...
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0answers
24 views

Understanding integration and substitution

I'm an undergraduate student in EE. I often see that when talking about voltages, curents ... being expressed like functions of some independed variable (time) and when calculating integrals people ...
0
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1answer
26 views

Empty function, what is it?

I meet with term 'empty function' from time to time. It's high time to understand its nature. What is field( set of arguments) and what is image? ( set of value)?
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0answers
9 views

general inverse of multivariable function

Given $m<n$. Let $f:N\subset\mathbb R^m\mapsto \mathbb R^n$ be a differentiable function. I am looking for the condition(s) such that we can find a function $g:im(f(N))\mapsto \mathbb R^m$ ...
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1answer
35 views

Who is growing faster?

I am trying to prove that $\lim_{n\to \infty} { 2^{n^2} \over n!} = \infty$. I can't use l'Hôpital's rule (or I dnon't know how) and I don't recall any other method which could help me. It also isn't ...
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2answers
27 views

Limit as x approaches 0 from the left: $\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$

Help me find the limit as x approaches 0 from the left: $$\lim_{x \to 0^{-}} \sin^{-1}\left({\frac{1}{2+e^\frac{1}{x}}}\right)$$ Thanks,
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1answer
11 views

Can the induced function of non-zero $f \in R[X]$ be zero, when $R$ is an infinite non-integral domain?

Let $R$ be an infinite commutative ring with $1$ which is not an integral domain. Is it possible to have a non-zero $f\in R[X]$ such that the induced map $\bar{f}: R \to R$ is zero? Please give ...
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1answer
33 views

Subspaces and annihilators

I am trying to show this question. My understanding of annihilators is that for a vector space $V$ over $K$, with $S$ being a subset, the annihilator of $S$ is the subspace $S^0$ of linear functions ...
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4answers
58 views

Find $\,\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$

How do I calculate $\lim_{x\to 1}\frac{\sqrt[n]{x}-1}{\sqrt[m]{x}-1}$. Please help me. Thanks!