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

learn more… | top users | synonyms (1)

0
votes
0answers
2 views

Determining concave, convex, quasi-concave and quasi-convex functions

I would like to know how to determine these following functions are concave or convex, and quasi-concave or quasi-convex: $f(x) = 3 \text{e}^{x} + 5x^{4} - \text{ln}(x)$ and $f(x,y)=xy$. Given the ...
0
votes
1answer
38 views

How to solve an equation in which x is an angle and we have x and cos(x)?

I have this function $$1.734\pi^2 *x /(980.553899* \sqrt{2*(1-cos(x)})$$ where x represents an angle in degrees and varies .01° to 90°. Can you please helpe understand: For x = 0.1° the ...
1
vote
1answer
40 views

injective and surjective

How to prove that a composite function $f\circ g$ is bijective$?$ because i have two questions. if $f$ is injective and $g$ is surjective, can $g\circ f$ be both injective and surjective? because the ...
1
vote
1answer
13 views

Is $f(g)$ homogeneous? If so, of what degree?

Given $f$ and $g$ are homogeneous functions of degree $k$. I have to show if $f(g)$ is homogeneous or not, and if so, of what degree. Definition (Homogeneous function). Let ...
0
votes
0answers
3 views

Laplace transform and inverse transform question

Appreciate advice on the above equation. I got the answer showed in the snapshot, but not sure whether it is correct or not.
1
vote
2answers
32 views

Showing that there exists a function with a certain property.

I want to prove the following: $\forall \epsilon > 0: \exists \text{ function }f:\mathbf{R}_{> 0} \rightarrow \mathbf{R}_{> 0}: \forall a,b\in \mathbf{R}_{>0}: a < b: \epsilon \cdot ...
1
vote
1answer
42 views

How do you prove that this function is bijective?

How do you prove that this function is bijective? $f\colon (0,1)\longrightarrow \mathbb{R}$ $f(x)=\tan (\pi(x-1/2))$ In fact I want to show that $(0,1)$ is equivalent to $\mathbb{R}$ by proving that ...
1
vote
1answer
15 views

Projection from triangle to spherical triangle

Consider a triangle, $T$, in $\mathbb{R}^3$ with vertices $(0,0,1), (0,1,0)$, and $(1,0,0)$. Let $S$ denote the sphere centered at the origin with radius 1 and let $S_1$ denote the portion of the ...
1
vote
2answers
97 views

Function or relation: y^4 = 8x^2

The question under debate: Is $y^4=8x^2$ a function or a relation? We disagree. I say, it's a relation because when I substitute in points I get two outputs for most inputs, and thus it's not a ...
-1
votes
0answers
24 views

Orthogonality of function spaces

I am not a mathematician, so sorry for the notation. I am working on finite element method/discontinuous Galerkin in one dimension. I would like to define function spaces $F_1, F_2, F_3$ with the ...
0
votes
0answers
41 views

If d/dx is an operation in functions, why do we need f(x)? [on hold]

This might be a little pedantic, but I need to sort out my terminology. Point 1:When mathematicians think of a function, they think of a mapping: $f:x \mapsto f(x)$. $f$ is a function that maps a ...
0
votes
1answer
39 views

How to find an inverse function of $\frac{2x+3}{x-1}$? [on hold]

How to find an inverse function of $\frac{2x+3}{x-1}$ step by step?
3
votes
2answers
49 views

Bijection between $\Bigl\{1, 2, \dots, \frac{N(N+1)}{2}\Bigr\}$ and $\{ (i, j) \in \mathbb{N} : i \le j \le N\}$

Let $N$ be some positive integer and $A$ be the following set $\{ (i, j) \in \mathbb{N}^2 : 1 \le i \le j \le N\} = \{ (1, 1), (1, 2), \ldots, (1, N), (2, 2), (2, 3), \ldots, (2, N), \ldots, (N, N) ...
0
votes
0answers
27 views

How can I extract value of s from Xi function?

How can I extract value of s from Xi function? $$\xi(s)=(s/2)\Gamma(s/2)(s-1) \pi^{s/2}\zeta(s)$$ Exmple $y=x^2-1$ $x=\sqrt {y-1}$ Yes, I mean inverse function.
0
votes
2answers
38 views

Finding the intervals where $f(x)=\frac{1}{|x-2|}-x$ is monotonous

Given $$f(x)=\frac{1}{|x-2|}-x.$$ I am interested in finding the intervals in $\mathbb{R}$ in which the function is monotonically increasing or decreasing. Usually I would take $f'(x)>0$ for the ...
0
votes
2answers
15 views

Finding all intervals in $\mathbb{R}$ for which the function is monotonically increasing/decreasing

Given $f(x) = x^{3}-3x+5$. How do I find all intervals for which the function is monotonically increasing and decreasing? I have $f'(x)=3x^{2}-3=0 \Rightarrow x=\pm 1$. And $f''(x)=6x$, so ...
0
votes
1answer
11 views

Investigating a function with parameters

I need a little help please, I got this function: $$y=\frac{x^2-1}{\sqrt{k-x^2}}$$ And this function has a vertical asymptote at $x=\sqrt 5$ How do I find $k$? Thanks :)
-1
votes
0answers
23 views

Finding domain of composition functions [on hold]

Q1.The functions f, g, h are given. Find formula for the composition $𝑓g,𝑔f ,ℎ𝑓,𝑓ℎ,ℎ𝑓g$. Write out the domain of each of the composite function: (1) $𝑓(𝑥)=1/2𝑥+1 ,𝑔(𝑥)=𝑥^3 ,ℎ(𝑥)=3𝑥+1.$ ...
2
votes
2answers
120 views

Basic composite function

If $f(x)={1\over x}$ and $g(x)=\sin(x)$ Just checking if I'm understanding this correctly. Are the formulas below correct? $$f(g(x))={1\over \sin(x)}\\ g(f(x))=\sin\left({1\over x}\right)$$ And ...
0
votes
0answers
12 views

Horizontal stretch factor: what is the correct way?

While at my summer school, I was taught that to translate a function, we should first isolate the x. Suppose the translated function is given by: $$ y = ...
2
votes
0answers
34 views

How do we address a function whose values are again functions?

One may call a function whose values are functions simply a function-valued function. But is there a canonical name for such an object? A $k$-form on some open $A \subset \mathbb{R}^{n}$ is an ...
0
votes
1answer
29 views

Derivative and and function terminology

In mathematical parlance, we say "take the derivative of a function f" to indicate that we are computing a new function, which maps slopes, that derives from f. However, in physics, we say "take the ...
2
votes
8answers
85 views

What is a surjective function?

I am a 9th grader self-studying about set theory and functions. I understood most basic concepts, but I didn't understand what is a surjective function. I have understood what is an injective ...
0
votes
1answer
21 views

Given a functional equation, its nature, value of its differential at a critical point, what are some methods to calculate its integral over a period?

My particular question is : If $f$ be a decreasing continuous function satisfying $$f(x+y)=f(x)+f(y)-f(x)f(y)$$$$ \forall x,y \in \mathbb{R}; f'(0)=-1$$ then $$\int_0^1f(x)dx =?$$ Answer to this ...
3
votes
0answers
16 views

Domain values of inverse funtion

If I'm plotting $$y=3e^{{x\over3}+1}$$ from $x=0$ to $x=1$ and on the same axes I want to plot its inverse $$y=3\ln\left({x\over3}\right)-3$$ but only for the domain values of $x$ given by the range ...
5
votes
1answer
27 views

Vector fields and tangent vector fields?

I am wondering if there are times when people would call a tangent vector field simply by a vector field? Are not these two concepts different? For example, a vector field assigns (say) to each ...
2
votes
3answers
115 views

Inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$

I am trying to find a proof of the following inequality $\left(\sum\limits_{k=0}^{2n-1}\frac{x^k}{k!}\right)\left(\sum\limits_{k=0}^{2n-1}\frac{(-x)^k}{k!}\right)\leq1$ for all $x\in\mathbb R$ and ...
-5
votes
0answers
31 views

Matrices and Linear Algebra- Determine if the list is linearly independent in the real vector space. [on hold]

1.Determine if the list $((3,2,0,1),\,(2,1,4,0),\,(0,-1,12,-2))$ is linearly independent in the real vector space $\mathbb R^4$. 2.In the real vector space $C(\mathbb R,\mathbb R)$ of all continuous ...
0
votes
1answer
12 views

show multivarable functions are one-to-one, onto.

$F:\mathbb{R}^3 \rightarrow \mathbb{R}^3, F(x,y,z)=(2x,y,3z+y)$ My current method for these sort of questions is to try to find the matrix that represents this transformation and then see if i can ...
0
votes
0answers
19 views

With 2 as smallest period of the function $f(x)$= $\tan^2[(\frac{\pi x}{n^2-5n+8})]$ + $\cot(n+m)\pi x$ ;the period m can't belong to is?

Here n $ \in N$ , m $\in Q$. Options are: A) $(-\infty, -2) \cup (-1, \infty)$ B) $(-\infty, -3) \cup (-2, \infty)$ C) $(-2,-1) \cup (-3,-2)$ D) $(-3, -5/2) \cup (-5/2, -2)$ I have an answer to ...
1
vote
1answer
24 views

Trying to construct a specific function

I am trying to construct a function $f$ with the following property: $\mathbf{N}$ is the set of natural numbers without 0. Show that $\forall \epsilon>0: \forall a,b \in \mathbf{N}: a < b: ...
-1
votes
0answers
20 views

function calculator [on hold]

I have a set of points and would like to know the function. x = 8 y = 338477 x = 12 y = 20484 I fail to see myself what the function could be. ...
1
vote
2answers
35 views

Problem with injective functions on an explanation of the Birthday problem

The Wikipedia article on the Birthday problem gives an "abstract proof" of the problem, in which the birthday function $$ b:\mathcal{S} \mapsto \mathcal{B} $$ where $\mathcal{S}$ is the set of ...
1
vote
2answers
45 views

Log function solve for x

The function is defined by $y=f(x)=3e^{{1\over3}x+1}$ Solve for $x$ in terms of $y$ My answer: $$x={\ln({y\over3})-1\over3}$$ Is this the correct way to go about this question? Update. Finding ...
0
votes
1answer
26 views

Consider a function $f(x)=1+2x+3x^2+4x^3$. Let $s$ be the sum of all real roots of $f(x)$ and $t=|s|$. Then…

the real number $s$ lies in the interval (A)$(-0.75,-0.5)$ (B)$(-0.5,0)$ (C)$(0,1)$ (D)$(-0.25,0)$ and the area of region bounded by $f(x),y=0,x=0$ lies in the interval (A)$(0.75,3)$ ...
0
votes
2answers
35 views

Is my explanation correct regarding Maximum value of Sine function over $\Bbb C$?

Question: What is the maximum value of sine function taking domain as $\Bbb C$? My answer is: The maximum value is not defined. Explanation: Since the range of sine function is $\Bbb C$ and $\Bbb C$ ...
0
votes
0answers
24 views

Is the function ${e^{-{1}\over{x}}}\over {x}$ on $(0,1)$ uniformly continuous or bounded?

$$f(x)= {{e^{-{1}\over{x}}}\over {x}}$$ for $x\in (0,1)$ . Is this function $a$) uniformly continuous $b$) bounded but not continuous $c$) unbounded This would be uniformly ...
0
votes
0answers
33 views

Maximum and minimum of $f(x)=x\sqrt{1+\sqrt{1-x^2}}+\sqrt{(1-x^2)(1-x)}$ when $-1\leq x\leq 1$

I am trying to find the maximum and minimum of $f(x)=x\sqrt{1+\sqrt{1-x^2}}+\sqrt{(1-x^2)(1-x)}$ when $-1\leq x\leq 1$. It seems that the minimum is $-1$, but I could not prove it. Anyway, does any ...
0
votes
2answers
34 views

Inverse functions: what is the difference between $\tan^{-1}(x)$ and $\tan(x)^{-1}$?

I’ve never really been taught about inverse functions, and I figured this is a pretty simple question, but I couldn’t find any explanation in my math textbook about this. What is the difference ...
1
vote
2answers
21 views

Composition and inverse mappings

Let $A\stackrel{\alpha} \rightarrow B \stackrel{\beta}\rightarrow A$ satisfy $\beta \alpha = 1_A$. If either $\alpha$ is onto or $\beta$ is one to one, show that each of them is invertible and that ...
1
vote
0answers
26 views

Abbreviating the definition of a tangent vector field?

Let $A \subset \mathbb{R}^{n}$ be open in $\mathbb{R}^{n}$ and let $F: A \to \mathbb{R}^{n} \times \mathbb{R}^{n}$ be continuous. Then $F$ is called a tangent vector field on $A$ if and only if $F(x) ...
0
votes
1answer
20 views

Approximation of 3d graph function

For a better figure I need to re-plot the following 3d graph: http://www.pic-upload.de/view-28191525/graph.jpg.html Could you tell me the rough function of it? Just ignore the red circle. I will ...
0
votes
1answer
14 views

How to calcuate the slope or gradeint usign function only with one point.

Hello I am studying functions deeply and I came up with a question. I have this function. f(x)=3x+2; I am reading a book. It says- The graph of f is a single line, passing through the point (0,2) ...
1
vote
1answer
24 views

Notation and name for this function?

Let $k \geq 1$; let $V,W$ be vector spaces; and let $T: V \to W$ be linear. Then how do we call and denote the function $(v_{1},\cdots, v_{k}) \mapsto (T(v_{1}), \cdots, T(v_{k})): V^{k} \to W^{k}$?
0
votes
3answers
28 views

Pre-images under the multiplication map $(x,y) \mapsto xy$ of open intervals are open

Let $f: \mathbb{R^2} \to \mathbb{R}$ be $f(x,y) = xy$. Find the pre-image $f^{-1}((a,b))$ of an open interval $(a, b) \subseteq \mathbb{R}$, and show that this pre-image is open in $\mathbb{R^2}$. ...
2
votes
4answers
39 views

Solving for b for equation $3a−5=−4b+1$ [on hold]

I have a math problem, and I'm trying to solve for 'b'. The problem answer shows that the first step is going from Step 1 to Step 2. I don't understand how they are doing this. How does the $-4b$ ...
0
votes
2answers
36 views

Domain and range of $f(x,y)=\sqrt{1+x-y^2}$

I need to find the domain and range of $f(x,y)=\sqrt{1+x-y^2}$. Can someone walk me through the proper reasonings in solving this problem? My attempt Domain From looking at the function I get: ...
0
votes
1answer
11 views

DNF or CNF functions

The problem tells us to find the full DNF and CNF of the logic function $f(P, Q, R)$ = True if and only if either Q is True or R is False. I feel fine with converting to get the full DNF or CNF form, ...
0
votes
0answers
13 views

Strict concavity when Hessian is only negative semi-definite for some values?

I am trying to find the values of a and b for which the function $f(x,y)=x^a+y^b$ is strictly concave over non-negative x and y. At first I was using the method of evaluating the definiteness of the ...
7
votes
1answer
71 views

Functional equation: Show $0\le f(n+1)-f(n)\le 1$ and find all $n$ such that $f(n)=1025$.

The function $f:\mathbb{N}\to \mathbb{R}$ satisfies all of $$\begin{align}f(1)&=1, \\ f(2)&=2,\\ f(n + 2) &= f(n + 2 − f(n + 1)) + f(n + 1 − f(n)) \tag{1} \end{align}$$ Show ...