Questions tagged [functions]
Elementary questions about functions, notation, properties, and operations such as function composition.
0
votes
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
votes
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
votes
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:
...
0
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-...
0
votes
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 ...
0
votes
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 ...
0
votes
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:
$$\cos2x=\frac{\...
0
votes
4answers
55 views
If $a+\sqrt{a^2+1}= b+\sqrt{b^2+1}$, then $a=b$ or not?
It might be a silly question but if $$a+\sqrt{a^2+1}= b+\sqrt{b^2+1},$$ then can I conclude that $a=b$? I thought about squaring both sides but I think it is wrong! Because radicals will not be ...
-1
votes
0answers
8 views
Economic applications of linear equations [on hold]
The profit of a firms product is linear and the marginal profit is 4. If the profit is equal to Ksh 100 when 102 units are sold.
Write down the equation of the profit function and determine the ...
0
votes
1answer
21 views
How to Prove The Complement Of The Domain Is Complement Of The Image If f Is Bijective
It seems true that $f(\overline{X}) = \overline{f(X)}$ for $f:A\rightarrow B$ and $X$ is any subset of $A$ if and only if $f$ is bijective.But I couldn't write it as a formal way like epsilon argument....
1
vote
2answers
40 views
Are functions $f(x)=\sin^2x-\sin^4x$, $g(x)=\cos^2x-\cos^4x$ equal or not? Why?
We have two functions named $f$ and $g$.
$$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)$$
Now the mission is to find out that they are equal functions or no.
...
-2
votes
2answers
24 views
Question of a infinitely differentiable function
I have tried using a few methods like Taylor series to solve the question but to no avail. Any help is appreciated.
2
votes
0answers
15 views
Price demand function to revenue function
I'm having a little bit of trouble figuring this out, I found the price demand function in the previous question, based on that I am supposed to find the revenue function. I am a bit confused by the ...
1
vote
3answers
32 views
If $(f\circ g)(x)=\tan^2x$ and $g(x)=\sqrt{\cos2x}$ then find $f(x)=?$
We've given :
$$(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})$$
But I do not know how to find $f(x)$...
0
votes
1answer
41 views
Monotonicity of function $f(x)=(1+\frac1x)^x(1+x)^\frac1x$
Given function for $x>0$
$$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$$
so, can we find a strict prove showing $f(x)$ ...
-2
votes
2answers
25 views
Bijection from [0,1] to [0,1] sum [2,3].
$0.0a_{2}a_{3}\ldots \rightarrow 0.a_{2}a_{3}\ldots$
$0.1a_{2}a_{3}\ldots \rightarrow 10.a_{2}a_{3}\ldots$
Numbers are base 2. Is this a valid bijection?
0
votes
1answer
13 views
Is $ \max_{x\in\mathbb{R}^n} \{ f(x)+g(x) \} = \max_{x\in\mathbb{R}^n} f(x)+\max_{x\in\mathbb{R}^n} g(x) $ if $f$ and $g$ are affine in $\mathbb{R}$?
Let $x \in \mathbb{R}^n$, and let $f(x)$ and $g(x)$ be two affine functions in $\mathbb{R}$.
Is the following property true?
$$
\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
$$f(x) =
\begin{cases}
x^2 - 1, & \text{if $x \leq 0$}\\
x - 2, & \text{if $0 < x < 1$}\\
c, & \text{if $x = 1$}\\
-x, & \text{if $x ...
0
votes
1answer
22 views
Arriving at odd possible solutions for functions
Say $f(x)=\dfrac{3x+1}{2}$
I want to find out for which values of $x$ is the value of $f(x)$ an odd number.
So I reframe $f(x)$ to
$\dfrac{3x+1}{2}=2k_1+1$
on simplifying further...
$\dfrac{3x-1}{...
4
votes
2answers
36 views
Can a continuous real valued function, differentiable everywhere but $x_0$, be expressed as $g(x)+h(x)|x-x_0|$ for some differentiable $g$ and $h$?
Let $f:\mathbb R \rightarrow \mathbb R$ be a continuous function and $x_0 \in \mathbb R$ such that f is differentiable on both intervals $(-\infty, x_0]$ and $[x_0, +\infty)$. Prove or disprove that ...
0
votes
1answer
26 views
Properties of Positive Real Functions
I am trying to understand the properties of positive real (PR) and strictly positive real (SPR) transfer functions. If given a transfer function I know how to determine whether or not the function is ...
0
votes
1answer
33 views
Finding a bijective rational function
I need to find a function that is bijective from $\mathbb{Q}$ onto $\mathbb{Q}^+$.
I was thinking of using $f(x)=2^x$ but does that satisfies the function being rational. I know the whole numbers are ...
0
votes
0answers
7 views
Is there a way to balance statistical measurements across differing total measurements?
Whew! Sorry for the terrible wording. My situation is this: I work in retail. A cornerstone of our performance resides in metrics that detail the amount of name and email information capture. ...
0
votes
3answers
39 views
Solving inverse function
$$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$$
...
0
votes
1answer
39 views
Why does this 2Pi-Periodic function looks like this?
I am given this function 1 on an interval $[0,\pi]$ and am asked to make a sketch of it on an interval $[-\pi,\pi]$. When plugging it into Maple, I am given a sketch graph like this 2, however, that ...
0
votes
1answer
17 views
Examples of functions characterized by sequence
Let $ \Psi$ be a set of functions $\chi :\mathbb{R}^+\rightarrow [0,1)$ satisfying
$$\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 ...
0
votes
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 ...
-2
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)$ ...?
-4
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
votes
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 ...
