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Questions tagged [functions]

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

0
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3answers
36 views

How do we write $f(x+2)$ in terms of $f(x)$?

$$f : R^+ \rightarrow R^+$$ $$f(x) = \dfrac{x}{x+1} $$ How do we write $f(x+2)$ in terms of $f(x)$? This is a general question that I wondered how to algebraically write it. Regards
-1
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2answers
17 views

If $g:\mathbb{Z_{10}}\rightarrow U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$.

Why is that if $g:\mathbb{Z_{10}}$$\rightarrow$$U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$? Also, $g$ is a function, $\mathbb{Z_{10}}$ is the group of integers ...
0
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1answer
30 views

How to find x0 ∈ (a, b) such that f(x0) ≤ M(b − a)/ 2 in this case?

Let $f(x)$ be a differential function on $[a, b]$ such that $f(a) = f(b) = 0$, $f(x) > 0$ where $x \in (a, b)$ and $f'(x)$ is continuous on $[a, b]$. (a) Prove that ∃M ∈ R such that |f 0 (x)| ≤ M ...
1
vote
0answers
29 views

For each interval $[a,b]$ contained in $I$, sequence $\{f_{n}:[a,b]\rightarrow\mathbb{R}\}$ converges uniformly to $f : [a,b] \rightarrow\mathbb{R}$

$I$ is an open interval Using the following fact to show this: ${\{f_n\}}$ converges pointwise on $I$ to the function $f$, and ${\{f'_n}\}$ converges uniformly on $I$ to the function $g$ Attempt: ...
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votes
1answer
24 views

Give an example of a function from $\mathbb N \to \mathbb N$ that satisfied

For each of the following properties give an example of a function from $\mathbb N \to \mathbb N$ that satisfied: (a) one-to-one but not onto (b) onto but not one-to-one (c) both onto and one-to-...
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1answer
24 views

How to show the existence of a root for a specific equation?

There is an interesting question about the solvability of the following equation. Let $a, b, c, d$ be constant numbers. In addition, these constant real numbers satisfy exponent $a >1$, finite ...
2
votes
1answer
34 views

Pigeonholes and onto functions

I've been scratching my head at this problem for a while and can't seem to figure out why the number of pigeonholes is $3^5 - C(3, 2)2^5 + C(3, 1)1^5$ and not $3^5 - C(3, 2)2^5 - C(3, 1)1^5$ ...
0
votes
1answer
11 views

formula for getting the normlized X and Y values of a given degrees from a linear function

I have a number which we'll call α in degrees that represent the angle of a linear function with the X axis. for example when α is 0 the linear function is on the X axis , when α is 360 the linear ...
1
vote
4answers
37 views

If $2f(x-1)+f(1-x)=2x$ then $f(x)=?$

We've been given a function: $$2f(x-1)+f(1-x)=2x$$ So now someone please teach me how to find: $$f(x)=???$$ Sorry! Really sorry but I don't have any ideas about it! Please tell me how can I solve this ...
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0answers
8 views

False negative and detection rate standard deviation values are always the same?

I am trying to recover standard deviation from the detection rate of an intrusion detection system through the false negative standard deviation. Until now I was able to recover the mean, maximum and ...
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2answers
23 views

How to rewrite $\cos2x$ in terms of $\frac{1-\sin x}{ 1+\sin x}$

Would you please tell me how to rewrite $\cos2x$ in terms of $$\frac{1-\sin(x)}{ 1+\sin(x)}$$ I had rewritten $\cos(2x)$ in terms of $\tan(x)$. But no results! I did this job! Here: $$\cos2x=\frac{\...
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4answers
55 views

If $a+\sqrt{a^2+1}= b+\sqrt{b^2+1}$, then $a=b$ or not?

It might be a silly question but if $$a+\sqrt{a^2+1}= b+\sqrt{b^2+1},$$ then can I conclude that $a=b$? I thought about squaring both sides but I think it is wrong! Because radicals will not be ...
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0answers
8 views

Economic applications of linear equations [on hold]

The profit of a firms product is linear and the marginal profit is 4. If the profit is equal to Ksh 100 when 102 units are sold. Write down the equation of the profit function and determine the ...
0
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1answer
21 views

How to Prove The Complement Of The Domain Is Complement Of The Image If f Is Bijective

It seems true that $f(\overline{X}) = \overline{f(X)}$ for $f:A\rightarrow B$ and $X$ is any subset of $A$ if and only if $f$ is bijective.But I couldn't write it as a formal way like epsilon argument....
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2answers
40 views

Are functions $f(x)=\sin^2x-\sin^4x$, $g(x)=\cos^2x-\cos^4x$ equal or not? Why?

We have two functions named $f$ and $g$. $$f(x)=\sin^2x-\sin^4x$$ $$g(x)=\cos^2x-\cos^4x$$ find out that they are equal functions or not. My problem is with: finding their domain if we choose any $x$ ...
1
vote
4answers
19 views

The function $g(x)=3x+\ln2x$ for $x>0$. Find $g'(x)$ and prove that $g(x)$ has an inverse function.

The function $g(x)=3x+\ln2x$ for $x>0$. Find $g'(x)$ and prove that $g(x)$ has an inverse function. So $g'(x)=3+\frac{1}{x} $ for $x>0$ But I have no idea how I'm supposed to prove that this ...
-1
votes
1answer
21 views

Are $f(x)=\operatorname{sgn}(x-\lfloor{x}\rfloor)$ and $g(x)=\operatorname{sgn}(\sin^2\pi x)$ equal functions? [on hold]

We have two functions named $f$ and $g$. $$f(x)=\operatorname{sgn}(x-\lfloor{x}\rfloor)\qquad g(x)=\operatorname{sgn}(\sin^2\pi x)$$ Now the mission is to find out that they are equal functions or no. ...
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2answers
24 views

Question of a infinitely differentiable function

I have tried using a few methods like Taylor series to solve the question but to no avail. Any help is appreciated.
2
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0answers
15 views

Price demand function to revenue function

I'm having a little bit of trouble figuring this out, I found the price demand function in the previous question, based on that I am supposed to find the revenue function. I am a bit confused by the ...
1
vote
3answers
32 views

If $(f\circ g)(x)=\tan^2x$ and $g(x)=\sqrt{\cos2x}$ then find $f(x)=?$

We've given : $$(f\circ g)(x)=\tan^2x$$ and $$g(x)=\sqrt{\cos 2x}$$ Then how to find the function $f(x)$? I know that $$(f\circ g)(x)=f(g(x))= f( \sqrt{\cos2x})$$ But I do not know how to find $f(x)$...
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1answer
41 views

Monotonicity of function $f(x)=(1+\frac1x)^x(1+x)^\frac1x$

Given function for $x>0$ $$f(x)=(1+\frac1x)^x(1+x)^\frac1x$$ which is not a monotonic function, but it is easy to find the only maxima $$f(1)=4$$ so, can we find a strict prove showing $f(x)$ ...
-2
votes
2answers
25 views

Bijection from [0,1] to [0,1] sum [2,3].

$0.0a_{2}a_{3}\ldots \rightarrow 0.a_{2}a_{3}\ldots$ $0.1a_{2}a_{3}\ldots \rightarrow 10.a_{2}a_{3}\ldots$ Numbers are base 2. Is this a valid bijection?
0
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1answer
13 views

Is $ \max_{x\in\mathbb{R}^n} \{ f(x)+g(x) \} = \max_{x\in\mathbb{R}^n} f(x)+\max_{x\in\mathbb{R}^n} g(x) $ if $f$ and $g$ are affine in $\mathbb{R}$?

Let $x \in \mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $\mathbb{R}$. Is the following property true? $$ \max_{x\in\mathbb{R}^n} \{ f(x) + g(x) \} = \max_{x\in\mathbb{R}^n} f(x)...
2
votes
2answers
18 views

Find the largest possible domain and the largest possible range of $F(x)$

(a) Let $F(x)=1+\cos2x$ . Find the largest possible domain and the largest possible range of $F(x)$. (b) $G(x)=x^2+2x-2, \;x \in [0, \infty)$. Find the inverse function $G^{-1}(x)$ and state its ...
-1
votes
1answer
17 views

At what point does both right hand and left hand limits exist, but the limit does not exist? Give your reason.

Question 1 Consider the function defined by $$f(x) = \begin{cases} x^2 - 1, & \text{if $x \leq 0$}\\ x - 2, & \text{if $0 < x < 1$}\\ c, & \text{if $x = 1$}\\ -x, & \text{if $x ...
0
votes
1answer
22 views

Arriving at odd possible solutions for functions

Say $f(x)=\dfrac{3x+1}{2}$ I want to find out for which values of $x$ is the value of $f(x)$ an odd number. So I reframe $f(x)$ to $\dfrac{3x+1}{2}=2k_1+1$ on simplifying further... $\dfrac{3x-1}{...
4
votes
2answers
36 views

Can a continuous real valued function, differentiable everywhere but $x_0$, be expressed as $g(x)+h(x)|x-x_0|$ for some differentiable $g$ and $h$?

Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous function and $x_0 \in \mathbb R$ such that f is differentiable on both intervals $(-\infty, x_0]$ and $[x_0, +\infty)$. Prove or disprove that ...
0
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1answer
26 views

Properties of Positive Real Functions

I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is ...
0
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1answer
33 views

Finding a bijective rational function

I need to find a function that is bijective from $\mathbb{Q}$ onto $\mathbb{Q}^+$. I was thinking of using $f(x)=2^x$ but does that satisfies the function being rational. I know the whole numbers are ...
0
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0answers
7 views

Is there a way to balance statistical measurements across differing total measurements?

Whew! Sorry for the terrible wording. My situation is this: I work in retail. A cornerstone of our performance resides in metrics that detail the amount of name and email information capture. ...
0
votes
3answers
39 views

Solving inverse function

$$f : \Bbb R \rightarrow \Bbb R $$ $$f^{-1}(2x-7) = x-1, f(a-1) = 5$$ Determine $a$. The inverse of the function $f^{-1}(2x-7)$ is written as $$f\biggr (\dfrac{x+7}{2}\biggr ) = x-1$$ ...
0
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1answer
39 views

Why does this 2Pi-Periodic function looks like this?

I am given this function 1 on an interval $[0,\pi]$ and am asked to make a sketch of it on an interval $[-\pi,\pi]$. When plugging it into Maple, I am given a sketch graph like this 2, however, that ...
0
votes
1answer
17 views

Examples of functions characterized by sequence

Let $ \Psi$ be a set of functions $\chi :\mathbb{R}^+\rightarrow [0,1)$ satisfying $$\chi(t_n)\rightarrow 0\Rightarrow t_n\rightarrow 1$$ I want to find some functions that belongs to $ \Psi$, other ...
1
vote
1answer
41 views

Is there a standard notation for a function's domain, codomain, and graph?

If I have a given function $f$, is there a standard notation for $f$'s domain, codomain and graph? One that I can use even if $f$ was not explicitly defined? The definition of a function, as I was ...
0
votes
1answer
22 views

How do you prove the operator of a function is also a function?

Let: $f:A\rightarrow B$ and $g: C \rightarrow D$ be functions. The "product of" $f$ and $g$ is the function defined as follows: $[f \cdot g](x,y) = (f(x),g(y)),\forall(x,y) \in A\times C$ How do you ...
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2answers
32 views

What is the concept of pathological function? [on hold]

mixed derivative theorem: Mixed partial derivatives Fxy and Fyx are always equal except for pathological functions. for using mixed derivative theorem function must be non-pathological,so i want a ...
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votes
3answers
59 views

$f(2x)=f(x+y)f(x-y)$ for all $x,y \in \mathbb{R}$

$f(2x)=f(x+y)f(x-y)$ for all $x,y \in \mathbb{R}$, and we also know $f(10)=4, f(0)\neq 0$, and $f'(0)=2$. Then what is $f'(10)$ ...?
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votes
2answers
36 views

If $f(x+y)=f(x)\cdot f(y)$ and $f(0)!=1$, prove that $f(0)=1$ [on hold]

I have only come this far: x+y=0 x=-y $f(0)=f(x)f(-x)=1 $ I can't go any further.
0
votes
1answer
17 views

How do I determine whether a function/map T is one-to-one and/or onto?

Given $T(x,y,z) = (xy, yz, xz)$, determine whether $T$ is one to one and/or onto. So far, I have come up with a contradiction that proves $T$ is not one to one: $T(-1, 1, -1) = (-1, -1, 1)$ and, $T(...
2
votes
1answer
25 views

Inverse Fourier transform with a $\delta$-function integrand

I'm self-studying math and trying to find the inverse Fourier transform of $\frac{4+w^2}{1+w^2}(4\pi * (\delta(w-2)+\delta(w+2)))$ Based on wolframalpha, the result is $32/5\sqrt{2\pi}\cos(2t)$. But ...
0
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0answers
17 views

What do we call function which satistfies $f(x,y,t) \le \vert M(1+\vert x \vert +\vert y \vert)\vert$?

What do we call function which satisfy $$ f(x,y,t) \le \vert M(1+\vert x \vert +\vert y \vert)\vert \text{for all } t \ge 0 \text{and } x,y \in \mathbb{R} $$ (are they sublinear?) and what about ...
-1
votes
4answers
37 views

Evaluate tan(x +y), and determine the quadrant it is located in.

If $x$ is in Q1 and $y$ is in Q2, $\sin x = \frac{24}{25}$ and $\sin y = \frac45$, evaluate $\tan(x + y)$, and determine the quadrant in which $x + y$ is located.
-4
votes
1answer
30 views

Derivative of Inverse trigonometric functions [on hold]

Suppose $g$ is the inverse function of a differentiable function $f$ and $G(x) = 1/g(x)$. If $f(3)= 7$ and $f '(x) = 1/9$, find $G\ '(7)$.
2
votes
3answers
69 views

Find the inverse of $f(x)=x^3+x+1$.

$$f(x)=x^3+x+1$$ I didn't learn this at school and I want to know how I can get the inverse of this function. Do you use differentiation? I have this solution but I don't understand what it ...
0
votes
0answers
19 views

When can I say my 2D mapping is really 1D?

Consider the following mapping: $$\left(\begin{matrix}x\\ y \end{matrix}\right)\mapsto\left(\begin{matrix}x+y\\ \alpha y+\sin\left(2\pi\left(x+y\right)\right) \end{matrix}\right)$$ with $x,y\in\...
0
votes
0answers
27 views

Domain of a function given only its formula?

Given an equation $f(x)=\cdots$, is there an accepted convention for defining the domain and range/codomain of $f$ (assuming these are not given). e.g. if $f(x)=x^2$ what is the domain and range of $...
0
votes
0answers
25 views

Finding first partial derivatives of a function?

So I have this function $f(x,y,z)=x\sin(y-z).$ And I want to find : the derivative with respect to $x$ the derivative with respect to $y$ the derivative with respect to $z .$ I think I should ...
0
votes
0answers
44 views

Proof for discontinuity in $f(x)$ at $x=0$ if $f(xy)=f(x)f(y)$ and $f(x)$ is continuous at $x=1$ [on hold]

My attempt: I found the left hand limit and right hand limit at $x=1$ and equated them From that it can be written that $\forall x,y\in \mathbb R, f(x)$ is continuous I don't see how $f$ can be ...
2
votes
2answers
75 views

let $A=\{1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5\}$. Number of one-one function from $A$ to $B$

Let $A=\{1,2,3,4,5\}$ and $B=\{0,1,2,3,4,5\}$. Number of one-one function from $A$ to $B$ such that $f(1) \neq 0$ and $f(i)\neq i$ for $i={1,2,3,4,5}$ is _______ . So I know one one function means ...
0
votes
5answers
56 views

Find the solution to $x^{(\log_5 x^2 + \log _5 x-12)}=\frac{1}{x^4}$

Find the solution to $x^{(\log_5 x^2 + \log _5 x-12)}=\frac{1}{x^4}$ I equated their exponents, That gave me $\log_5 x = \frac{8}{3}$ But the answer given in my book is $1$. Obviously, 1 satisfies ...