# Tagged Questions

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

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### Determining bounds for a sum with nested infinite series

I am computing the inner product of the characters of the trivial and the $k$-th irreducible two dimensional representations of the dihedral group $D_n$ of order $2 n$ when $n$ is even. The ...
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### How to show that the following function is bijective?

If we have the function $c : \mathbb{N}^2 \rightarrow \mathbb{N} : (x,y) \rightarrow 2^x \cdot (2y+1) -1$ how to show that this function is bijective? So I thought the easiest way is to show that is ...
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### Find the Domain of the Function g such that gf and g inverse exists.

The functions $f$ and $g$ are defined as follows: $$f(x)=(2e^x-1)^2+2, x\in\mathbb R$$ $$g(x)=(x-1)^2-1, x\ge k$$ Find the range of values of $k$ such that both functions $gf$ and $g^{-1}$ exists. ...
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if the equation $$(x-2)e^x+a(x-1)^2=0,x\in R$$ have two real roots. show that $$a>0$$ Following is a solution since $$-a=\dfrac{(x-2)e^x}{(x-1)^2}$$ Let $$g(x)=\dfrac{(x-2)e^x}{(x-1)^2}\... 1answer 49 views ### Show that … Has No Real Roots When f(x) = 3x-4 and g(x) = \frac{5}{3-x}, Question 1: Find the value of x for which fg(x) = 5 Question 2: Show that the equation f^{-1}(x) = g^{-1}(x) has no real roots. I understand that ... 3answers 78 views ### Proving that a function is surjective I want to prove that the function \mathbb{N}_0 \times \mathbb{N}_0 \rightarrow \mathbb{N}_0 defined as (x,y) \mapsto 2^x \cdot (2y + 1) - 1 is bijective. I have already proven that it is injective,... 1answer 14 views ### How to design a threshold function without using any comparison operator? What are some methods to design a function that outputs 1 if the input value x is greater than a threshold T and 0 otherwise. f(x,T)=\begin{cases} 1,x\geq T\\ 0, x<T \end{cases} 1answer 53 views ### Is it possible to construct such a function in analytical form? Suppose f\left(f\left(x\right)\right)=\sin(x) Is it possible to find f in closed form, or any other forms so as to visualize f(x) on x\in[-\pi,\pi]? Is it possible to prove the existence and ... 1answer 36 views ### Graph the function [closed] Graph the function$$ f(x)=\begin{cases} -x-2, & -2<x\le -1 \\ -x^2, & -1<x\le 1 \\ x+2, & 1<x\le 2 \end{cases} $$1answer 32 views ### Precalculus questions: Domain, range, and composition of functions Directions: evaluate each of the functions at the indicated value of x. construct each of the functions, then find the domain. If f(x)=\{(3,5),(2,4),(1,7)\}, g(x)=\sqrt{x-3}, h(x)=\{(3,2),(4,3)... 1answer 37 views ### Proof of a Surjective Function I've run into a question in my textbook and I'm not sure if I understand fully the answer from the solution manual. Here is the question: Problem: Suppose that f: A \rightarrow B is any function. ... 2answers 50 views ### Find a function that is a bijection f:(0,1) \rightarrow (1, \infty) Find a function that is a bijection f:(0,1) \rightarrow (1, \infty) I am to assume the intervals have the same cardinality. I honestly don't even know how to begin with this. Can you provide me ... 0answers 17 views ### Image maps for Relations So I noticed a rather interesting thing about the image function and preimage function. The wikipedia page for Image: https://en.wikipedia.org/wiki/Image_(mathematics) (notation for image section) ... 3answers 71 views ### How is the function f: \mathbb{Z} \to \mathbb{R} continuous? Where \mathbb{Z} is the set of integers and \mathbb{R} the set of real numbers. In a question in a problem sheet, it said this statement was correct, however I do not understand how. You ... 2answers 23 views ### Description of preimage of g defined by g|_A = f|_A, g(S \setminus A) = \{x\} S is a given directed set. X is a first countable topological space. There is a function f : S\rightarrow X. With help of this f , a new function g:S\rightarrow X is being defined like the ... 0answers 28 views ### How do I determine Taylor polynomial of degree 2 of function? How do I determine Taylor polynomial of degree 2 of function g(x,y) = (x^2 + y)e^{xy} in development at point (x_0,y_0)=(1,-2)? There is this question, but the problem is that I need to do it on ... 1answer 22 views ### How do I determine set and sketch image of function g(\mathbb{R}^2) if g(x,y)=\begin{pmatrix} e^x\cos y\\ e^x\sin y\end{pmatrix}? How do I determine set and sketch image of function g(\mathbb{R}^2) if g(x,y)=\begin{pmatrix} e^x\cos y\\ e^x\sin y\end{pmatrix}? 1answer 34 views ### Find functions F(\mathbf{x}) invariant under a map \mathbf{x} \to \mathbf{x'} We introduce a map \mathbf{x} \to \mathbf{x'}, defined as (for example on \mathbb{R}^3):$$x'=f(x,y,z) \\ y'=g(x,y,z) \\ z'=h(x,y,z)$$Note that f,g,h are not all linear (or at least, I'm not ... 1answer 36 views ### How to solve problem like this - even & odd function This is my problem, Consider f(x)=\frac{1}{e^{x}} , Where x\epsilon [0,\infty[ 1) Define the function g:\Re \rightarrow \Re such that g is an even function and g=f on [0,\infty[ Some ... 0answers 26 views ### Inequality between norm of function and it's derivative There is a theorem: Let f be a continuously differentiable, 2\pi-periodic function. Given \int_{-\pi}^{\pi} f(x) dx = 0, I need to prove that$$||f|| \le \frac{\pi}{2} \cdot ||f'||.$$Where ... 1answer 44 views ### How to sketch an image and find set of this function?  c\left(t\right) =\begin{pmatrix} e^{t} \\ e^{-t} \end{pmatrix}  How do I find set and geometric object of  c\left(\mathbb R \right)  ? I know the graph of e^{t} and e^{-t}, but how to put ... 1answer 26 views ### How do I determine some function if domain and range are already specified? In one of my college interviews, I was asked to define a function whose domain is (0,1) and range is (-1,1). This can be very well answered (we can modify 'x' in cos(x)). I did encounter certain ... 3answers 30 views ### reciprocal factor of absolute value when evaluating a square root expression Learning with an old russian math book, i found the following evaluation for the function f(x)=\sqrt{1+x^2}: f(\frac1x)=\vert x \vert^{-1}\sqrt{1+x^2} My evaluation gave me \sqrt{1+\dfrac1{x^... 0answers 28 views ### Function which gives number of primes STRICTLY less than a number x. We all know that the \pi(x) function (Prime-counting function) gives us the number of primes less than or equal to x. I'm interested in the function which specifically gives the number of primes ... 1answer 21 views ### Proving the complete additivity of the Big Omega function \Omega(n) (total number of prime factors of n) . The Big Omega function \Omega(n) gives you the total number of prime factors of the number n. A function f(x) is completely additive if f(ab)=f(a)+f(b) for all positive numbers a and b, ... 0answers 27 views ### Find the sets for f:R \rightarrow R given by f(x) = \left|x\right| Find the sets for f:R \rightarrow R given by f(x) = \left|x\right|. Let S= [0,4] and T = [-3,0]. Find the sets: f(S), f(T), f(S) \cap f(T), and f(S \cap T). Is f(S) \cap f(T) = f(... 2answers 52 views ### How do I determine the maximum or the minimum of in the range the function in the range B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}? How do I determine the maximum or the minimum of in the range the function f(x,y)=x^3+y^3 in the range B=\left \{ \left ( x,y \right ):x^2+y^2\leq 1 \right \}? As a continuous function f must ... 0answers 21 views ### Symmetry vs. commutativity and more.. I was reading the following segment of an article about commutativity: ... 0answers 42 views ### My proof that f^{-1}(D_1 \cap D_2) = f^{-1}(D_1) \cap f^{-1}(D_2) I'm currently self studying proof and set-theory, and I'm quite new to both of them. As an exercise, I'm practicing proving some basic theorems, so it'll be great if you can give me some feedback on ... 3answers 28 views ### Find the inverse of f(x) = 1 + \frac{1}{x}, x \gt 0 I'm tasked to find the inverse of the function$$f(x) = 1 + \frac{1}{x}, x \gt 0$$The book offers a solution, simply to set$$1 + \frac{1}{x} = s$$and solve$$x = \frac{1}{s-1}$$and I think I ... 1answer 44 views ### Injectivity in functions Sorry, I know that it has to be a very simple problem, but I'm frustrated because of it. Let f,g:\mathbb{N}^3→\mathbb{N}f: f(x,y,z)=3^x⋅5^y⋅7^z and g(x,y,z)=3^x+5^y+7^z Prove that: 1.f is ... 1answer 29 views ### c_1\cosh(x)+c_2\sinh(x)=A\cosh(x+y) always true? My question: Can I rewrite c_1\cosh(x)+c_2\sinh(x), which is a solution to a differential equation as$$A\cosh(x+x_0)$$introducing the new constants of integration A and x_0? How can I deal ... 1answer 39 views ### Determine and sketch the image of function? I have to sketch the image of function h\left(\mathbb R^{2} \right) as 1. a set and as 2. a geometric object. h\left(r,\phi\right)=\begin{pmatrix} rcos\phi \\ rsin\phi \\ r \end{pmatrix}  I ... 2answers 68 views ### Finding all possible values of a Function Let a function be defined as f:N\to N and x-f(x)= 19\left[\dfrac{x}{19}\right] - 90\left[\dfrac{f(x)}{90}\right] \forall x\in \Bbb N and 1900<f(1990)<2000. Find all values of f(1990). ... 0answers 42 views ### Surface area of a bottle with integration Would it be possible to model a bottle using a function, then revolving it to determine the surface area and the volume while customizing the curvature and the dimensions of particular sections of the ... 0answers 15 views ### show that Lb(a, y) := max{1−ay−by,0}, (a, y) ∈ R×{−1,1}, b ∈ R, is continuouse whit respect to first variable Show that:$$L_b(a, y) = \max\{1−ay−by,0\},\;\; (a, y) \in\mathbb{R}×\{−1,1\}, b\in\mathbb{R} is continuous whit respect to first variable.
Let $f(x)$ is a polynomial satisfying $f(x).f(y)=f(x) + f(y) +f(xy) -2$ for all x ,y and $f(2)=1025$ , then the value of lim x tending to 2 $f'(x)$ is I want to know that value at $f(1)=1$ can ...