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

learn more… | top users | synonyms (1)

1
vote
0answers
47 views

Are there such prime giving functions?

Here let us define a function $f : \mathbb{N} \rightarrow \mathbb{N}$ , such that for every $n$ , The sequence $\{f(n) ,f(n)+1 ,f(n)+2 , f(n)+3, \dots , f(n)+n\}$ contains atleast $1$ prime . Let us ...
2
votes
2answers
64 views

Find all holomorphic functions on $\mathbb{C}$, except for some singularities, such that $|f(z)|\leq C(|z|^{3/2}+|z-1|^{-3/2}), z\in\mathbb{C}-\{1\}$

First I wrote the Laurent series of $f(z)$ around $z=1$: $$ f(z)=\sum_{n=-\infty}^{-1}c_n(z-1)^n+\sum_{n=0}^{\infty}c_n(z-1)^n. $$ Now if $|z|$ becomes very large, the first sum with the negatives ...
0
votes
1answer
57 views

Show that $f(0)=1$ given that $f(x+y)=f(x)f(y)\:\;\forall x,y\in\Bbb R$.

The question is Let $f$ be a function with domain $\Bbb R$ that satisfies the conditions, $$f(x+y)=f(x)f(y),\;\text{for all $x$ and $y$ and $f(0)\neq0.$}$$ $(a)$ Show that $f(0)=1$. The mark ...
1
vote
1answer
20 views

Show that the following functi0n is bounded

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function with $f(0)>0$ and $$\lim_{x \to \infty} f(x)= \lim_{x \to -\infty} f(x)=0$$ $(i)$ Show that $f$ is bounded. $(ii)$ Let ...
0
votes
2answers
718 views

Find cubic equation given four points?

I am working on finding the area of a solid object. I have 4 points that I need to calculate a cubic equation from. I have tried relentlessly but to no avail I always get the wrong answer. The four ...
1
vote
1answer
35 views

Kalman's controllability condition implies that the r.h.s. is “non-degenerate.”

Assume $f:\mathbb R^n\times\mathbb R\to\mathbb R^n$ is given by $f(x,u)=Ax+bu$ for some choice of coordinates $(x,u):x\in\mathbb R^n,u\in\mathbb R$, some constant matrix $A\in M_n(\mathbb R)$, and a ...
0
votes
2answers
33 views

Quadratic equation problem solving

Below are two functions. These functions were both originally the same function $f(x)$ but were transformed using the constant 'p' by either $f(x)+p, f(x+p), p \cdot f(x)$ or $f(p \cdot x) $ If p=3 ...
0
votes
1answer
25 views

Finding rate in exponential decay

Using the exponential decay eqution: I = Io * e^(-kx) -k = rate, x = time, Io = initial amount I was asked to find the rate (-k). We were given the following information, when x = 2 I = 12 and when ...
1
vote
1answer
34 views

Proof that the exists a bijective funtion

$S$ be a set. Consider the set of all functions from $S$ into $\{0,1\}$. The set is $2^S$ How do I proof that there exists a bijective function from $P(S)$ to $2^S$
0
votes
2answers
31 views

Is the function bijective?

let $B$ be the set of all binary strings over the alphabet $\{0,1\}$. Consider the function $f \colon B\to B$ such that for any string $x$, the value $f(x)$ is obtained by replacing all $0$'s in $x$ ...
0
votes
1answer
40 views

Calculus-Tangent Line

Find the cordinates of the point on the curve $f(x)=xe^{2x}+1$ where the tangent of the tangent line is horizontal. I am not sure of what to do.
2
votes
5answers
2k views

Why is this function neither injective nor surjective?

Hello guys I am doing some maths revision and I am stuck. I read and I saw videos and I still can't get my head around it. I have this example: Determine whenever this function ...
0
votes
2answers
49 views

Inverse functions problem

There are two functions $f\colon\mathbb Q \to \mathbb Q \setminus \{-1\}$ and $g\colon\mathbb Q \to \mathbb Q \setminus \{1\}$. $$g(x) = \frac{f(x)}{f(x)+1}.$$ Prove that if there is a inverse ...
0
votes
1answer
47 views

Proving that a function is periodic

I need help proving the following: Let $f(x)$ be an even function and let $A$ be an arbitrary real number . If the function $g(x) = f(A - x) $ is odd then $f(x)$ is periodic.
1
vote
2answers
27 views

Name for a set of pairs of elements that equalise two functions?

Is there an established name for this $eql$ function? $$\operatorname{eql}(f, g) = \{\ (x, y)\mid f(x) = g(y)\ \}$$
2
votes
2answers
102 views

Prove that f is differentiable at $0$! Not continuous though, Right!?

Suppose $f(x)$ equals $x^2$ when $x\in \mathbb{Q}$ and $0$ when $x \not\in \mathbb{Q}$ Prove that $f$ is differentiable at $0$ and find the derivative $f'(0)$ Shouldn't this be obvious, since $x^2$ ...
2
votes
1answer
46 views

how to prove the following relation

Let $f$ be a continuous function on [0,1] differentiable on (0,1) such that $f(1)=0$ then prove that for some $c$ $$cf'(c)+ f(c)=0$$
0
votes
2answers
29 views

combining ratios

I am trying to work out how to blend compounds. Compound 1 has 60% A and 15% B and 25% C Compound 2 has 70% A and 5% B and 25% C I want to make a new compound with 75% A and 10% B and 25% C How ...
0
votes
0answers
59 views

Is $2\delta(x) \neq \delta(x)$?

$2\delta(x) \neq \delta(x)$ since, by definition, Can this been seen graphically though? If so, how? If not, why is it that they are mathematically different but graphically the same? Btw, I ...
2
votes
2answers
40 views

The domain of $f(x)=\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}$

$$f(x)=\sqrt{\frac{1-\cos(x)}{1+\cos(x)}}$$ It's obvious that this: $$\frac{1-\cos(x)}{1+\cos(x)}\geq0$$ and this $$1+\cos (x)\ne0$$ $$x\ne\pi+2k\pi, k\in Z$$ are the conditions of the function's ...
2
votes
1answer
25 views

Given three continuous and diffrentiable functions satisfying given condition, which of the following options are correct

Given, $f(x),g(x),h(x)$ are three continuous and differentiable functions on $\Bbb{R}$ Which of the following statements are true if $$f(x+y)=g(x)+h(y)\qquad\forall(x,y)\in\Bbb{R}$$ a) $f'(0)=f'(1)$ ...
2
votes
1answer
74 views

Does the function differentiable?

Let $\alpha, \beta \ge 1 \in \mathbb{N}$ and: $$f(x) = \left\{ {\matrix{ {{x^\beta }\sin \left( {{1 \over {{x^\alpha }}}} \right)} \cr {0,x = 0} \cr } } \right.$$ I checked for ...
3
votes
2answers
26 views

This function is injective

I'm trying to prove this function is injective: $$f:P(\mathbb N)\to \mathbb R, f(M)=\sum_{n\in M}3^{-n}$$ I've already proved that this function is well-defined but I couldn't prove this function is ...
0
votes
1answer
52 views

Inverse image of an element in co-domain but not in range?

Sorry, quite new to this. I have a question that contains the image below of $g:X\rightarrow Y$ and it is asking for the inverse image of $u$. Am I correct in thinking that the answer is $\emptyset$? ...
6
votes
1answer
194 views

A counterexample

Let $f:\mathbb{R}\to(0,\infty)$ a locally integrable function. I want to compare these two conditions $$\limsup_{r\to + \infty}\frac{r}{\int_{-r}^r f(x)dx}<+\infty. \tag{1}\label{1}$$ and ...
1
vote
0answers
19 views

unimodality and continuous

i would like to ask question about unimodality of probability function ,from wikipedia http://en.wikipedia.org/wiki/Unimodal it says that In mathematics, unimodality means possessing a unique mode. ...
1
vote
2answers
52 views

f(a) = b, a to b or b to a?

I'm reading a book, I quote from it: "" A function assigns an element of one set, called the domain, to elements of another set, called the codomain. The notation $f: A \to B$ indicates that $f$ is a ...
2
votes
2answers
98 views

Functional inequation on $\mathbb{R}$: $f(x+y^2)-f(x)\geq y$

I have the following equation: $$f:\mathbb{R}\rightarrow\mathbb{R}$$ $$\forall (x,y)\in\mathbb{R}^2,\ f(x+y^2)-f(x)\geq y$$ f is not necessarily differentiable/continuous/... (In fact, we can prove ...
4
votes
1answer
132 views

Is $f(x)=\log(1+x^2)$ uniformly continuous on $(0,\infty)$?

Is $f(x) = \log(1+x^2)$ uniformly continuous on $(0,\infty)$? My work: Looking at the graph and knowing that $\log$ considered a "slow-growing" function, my guess is that $f(x)$ is uniformly ...
1
vote
1answer
70 views

Discrete Math - onto, 1-1 functions

Let $S = \{3,B\}$ Give an example of a function $f: S \times S \to S$ that is onto. Give an example of a function $g: S \to S \times S$ that is 1-1. Give an example of a function $h: ...
0
votes
1answer
34 views

Can you find such a function that satisfies the RH statement?

For instance, see Generalized Riemann Hypothesis. It conjectures that if $L(\chi, s) = 0$, and $0 \leq \Re(s)\leq 1$, then $\Re(s) = 1/2$. Then is there a function $f(s)$ that you can think of that ...
2
votes
4answers
90 views

Finding the inverse of $f(x)=|x|-2$

How would I find the inverse of the function $f(x)=|x|-2$? I have swapped $x$ and $y$, and tried to isolate $y$, reaching up to $x+2=|y|$ Whenever I see absolute values, I always break the problem up ...
2
votes
1answer
51 views

Uniform distribtion: clarification of $f_X(x)$

I have $Y=2(X-1)^2 -1$ where $X$ is uniform distributed on $[0,2]$ I want to find the pdf of $Y$ and expected value of $Y$. My question is just: Does $X$ have pdf $f_X(x)= \frac{1}{2}$?
12
votes
1answer
151 views

Let $(f(x))^2$ and $(f(x))^3$ are $C^{\infty}$. Prove or disprove that $f$ is $C^{\infty}$.

Suppose $f(x), -\infty < x < +\infty$, is a real valued function such that both $(f(x))^2$ and $(f(x))^3$ are $C^{\infty}$. Must $f$ be $C^{\infty}$? I had seen this exercise ...
0
votes
4answers
42 views

Surjectivity of polynomials

Show that the mapping $f$:[$0,\infty$]$-R$ given $f(x)=x^2+2x-8$ for all $x$ belonging to [$0,\infty$) is invective but not surjective. How do you prove the surjectivity part? Because the general ...
0
votes
0answers
25 views

Surjective complex map.

Show the map $(a + bi) \mapsto (a-bi)$ is surjective. Attempt: By definition, for every $(a - bi)$ in the complex set, there exists an $(a + bi)$ in the complex set such that $f[(a + bi)] = a - bi$. ...
1
vote
1answer
67 views

Convergence and uniform convergence of a sequence of functions

Suppose that ($f_n$;$n \geq 1$) is a sequence of functions defined on an interval [a,b].We will say $f_n$ tends to $f$ uniformly on $[a,b]$ as n tends to infinity if for every $\epsilon>0$ there ...
1
vote
1answer
46 views

sequence of analytic functions on an open subset of $\mathbb{C}$ that converges uniformly on compact subsets

Let $U$ be an open subset of $\mathbb{C}$ and let $\{f_n\}$ be a sequence of analytic functions on $U$. Suppose that $(f_n)$ converges uniformly on any compact subsets of $U$ to a function $f$. Let ...
1
vote
2answers
38 views

Find the minimal value of a function

Say we have function: $ f(r) = \frac{b}{r} (n + 2^r), r > 0 $ where $b$ and $n$ are some constants large than $0$. How can we determine the minimal value of this function? Compute the ...
2
votes
4answers
45 views

Trying to understand how many functions there are from A to B.

I'm trying to understand why there are $B^A$ functions from $A$ to $B$. If $A=${$a,b$} and $B=${$1,2,3$} then the functions from $A$ to $B$ are $f(a)=1$, $f(b)=1$, $f(a)=2$, $f(b)=2$, $f(a)=3$, ...
3
votes
1answer
159 views

Finding the equation for a sinusoidal cycle/function given points.

We are given the population of a fictional animal at different years: $$\begin{array}{l|r} \textrm{Year} & \textrm{Population}\\\hline 1945 & 347,0000\\ 1955 & 76,000\\ 1965 & ...
1
vote
1answer
445 views

Find the linear-to-linear function whose graph passes through the given three points

Find the linear-to-linear function whose graph passes through the points $(1, 1)$, $(4, 2)$ and $(30, 3)$. So by using the $$f(x)=\frac{ax +b}{x+d}$$ I got my final answer to be ...
1
vote
2answers
83 views

Iterated self-composition of arbitrary function

Does there exist some notation that represents the iterative composition of a single-input, single-output function with itself? As in, say, $f_5(x)=f(f(f(f(f(x)))))$. In other words, going by the ...
2
votes
2answers
88 views

Functional equations problem 3

Find the functions that satisfy the relation $$f(x)f\left(\frac{1}{x}\right)=f(x)+f\left(\frac{1}{x}\right).$$ Did not get any idea how to do this.
2
votes
3answers
47 views

Name of function

I am sorry if this is a stupid question, but I am struggling to give the proper name to the following function: $$\ f(r) = \exp(f_1+f_2r+f_3r^2+f_4r^3+f_5r^4+f_6r^5)$$ I ask as it will be in a ...
1
vote
2answers
239 views

Find a function that satisfies the condition.

Let $\epsilon > 0$ be fixed and $t$ a variable that takes values in the universal covering space of ${\mathbb{C} \setminus \{0\}}$. Find a continuous function $f(s$) such that $$|t \log t| = |t| ...
7
votes
2answers
145 views

Continuous functions that satisfies $f(x) + f(1-x) + f(\sqrt{x^2+(1-x)}) = 0$ and $f(\frac12)=0$

$f:[0,1]\rightarrow \mathbb{R}$ is a continuous function which satisfies $$ f(x) + f(1-x) + f\left(\sqrt{x^2+(1-x)}\right) = 0 \text{ and } f\left(\tfrac12\right)=0. $$ Can someone give explicit ...
0
votes
3answers
24 views

Prove that the range of $f$ does not contain $1$.

Let $f(x) = (\ln ({7x−x^2\over12}))^{3\over2}$ Prove that the range of $f$ does not contain $1$. My approach: $f$ is defined for $x(x-7)<0$. So, range of $f$ should be $(0,7)$. $\therefore$ It ...
1
vote
1answer
34 views

$f(x) = e^{-{1\over x^2}}+\int_0^{\pi x\over2}(1+\sin t)^{1\over2}dt$ for $x\in(0,\infty)$

Let $$f(x) = e^{-{1\over x^2}}+\int_0^{\pi x\over2}(1+\sin t)^{1\over2}dt$$ for $x\in(0,\infty)$ Then which of the following are true? (A) $f′$ exists and is continuous. (B) $f′′$ exists ...
1
vote
1answer
43 views