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

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2
votes
5answers
57 views

Prove $\frac{2\cos x}{\cos 2x + 1 }= \sec x$

Prove that $\dfrac{2\cos x}{\cos 2x + 1 }= \sec x$. So far I have: $\dfrac{2\cos x}{\cos 2x + 1 }= \dfrac 1 {\cos x}$ Where do I go from here?
3
votes
3answers
99 views

If $a$ and $b$ are roots of $x^4+x^3-1=0$, $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$.

I have to prove that: If $a$ and $b$ are two roots of $x^4+x^3-1=0$, then $ab$ is a root of $x^6+x^4+x^3-x^2-1=0$. I tried this : $a$ and $b$ are root of $x^4+x^3-1=0$ means : $\begin{cases} ...
0
votes
1answer
21 views

When is the preimage of codomain not equal to domain?

I need to show that For every $X\subseteq A$, $X \subseteq f^{-1}(f(X))$, where $f: A\to B$. I think I understand why this is true. However, under what circumstances are they not equal? When is ...
-1
votes
0answers
45 views

Limit of difference between function and its limiting value [closed]

If we to find limit of function $[(π/2)-x]\tan x$ when $ x $ approaches to $π/2$ . I solved this by putting $x= π/2 + h$ and $x= π/2 -h$ where $ h $ is very small number Then I got the limiting ...
1
vote
0answers
23 views

Constructing an algebraic surface with singularities on the unit circle.

I am currently doing a project on algebraic surfaces, and I want to construct an algebraic surface $\mathbb{V}(f(x, y, z))$ that exhibit its singularities on the unit circle $x^2 + y^2 - 1 = 0$. My ...
0
votes
1answer
47 views

Quick clarification: Justification for properties implying the zero function

I came across a result that claims for a function $f(x)$ that $f(x + \frac{1}{n}) = f(x)$ $f(x) = 0, \hspace{2mm} \forall x \in [0, \frac{1}{n})$ then $f(x) = 0,$ $\forall x \in \mathbb{R}.$ Is ...
1
vote
1answer
33 views

Does the function assume its minimum value at only one point or not?

I am having trouble proving whether this statement is true or false.Can someone help me please.Thanks !
0
votes
2answers
43 views

Solve $f(4x-5)$ and $f(-7)$ when $f(2x+1) = x^2 +4x +2$

I know that if $f(x)= ...$ and when $f(-7)$ given, I just have to plug it in and solve. But for this it had given $f(2x+1)$, so can anybody guide me to this equation?
-1
votes
2answers
71 views

Proof without words for logarithmic funtions [closed]

I'm looking for a PROOF WITHOUT words for logarithms. The only one I've seen is calculus based and I need one for a younger audience. Any help/suggestions would be appreciated! This is the example I ...
0
votes
1answer
14 views

forming log equation from graph points

Okay so I need to form a logarithmic equation from the points (1960,4.7) (1964,5.1) (1968,5.4). I have 'guess and checked' to get the equation 2.7421 log(x-1950)+1.9579, and was wondering if there was ...
3
votes
1answer
50 views

fundamental period of function $f(x)$ is

If $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(2+x) = f(2-x)$ and $f(20-x) = f(x)\;\forall x\in \mathbb{R}$ and $f(2)\neq f(6)$ Then fundamental period of function $f(x)$ is $\bf{My\; ...
2
votes
1answer
33 views

Proving an identity involving floor function

Prove that : $$\left \lfloor \dfrac{2 a^2}{b} \right \rfloor - 2 \left \lfloor \dfrac{a^2}{b} \right \rfloor = \left \lfloor \dfrac{2 (a^2 \bmod b)}{b} \right \rfloor $$ Where $a$ ...
0
votes
2answers
24 views

Proving a function has a bijection

Can someone give me an example of a bijective function f: A → B with two finite sets? Suppose A = {1, 2, 3, 4} and B = {a, b, c, d}. Can you prove that it is injective and surjective?
1
vote
0answers
12 views

Proving alternate definitions of homogeneous functions are equivalent

The definition of a homogeneous function $f(x, y)$ of order $n$ that I'm used to is $$f(tx, ty) = t^n f(x, y)$$ for all $t > 0$. I recently encountered another definition: $f(x, y)$ is homogeneous ...
-1
votes
2answers
45 views

Prove that $f \circ g = g \circ f$ from A to A [duplicate]

How do I prove that if A is set and each of f and g is a function from A to A, then f o g = g o f? Edit: If this is not true how can I prove that it is false using sets?
0
votes
2answers
49 views

Proving a function is bijective

Suppose set $A =\{1, 2, 3, 4\}$ and set $B = \{w, x, y, z\}$. Prove that the function, $f: A \to B$ is both injective and surjective (bijective). Use any function.
2
votes
3answers
355 views

WolframAlpha function that returns the 'decimal' part of a number [closed]

Is there a function or command in Wolfram Alpha for getting only the decimal part of a number? Something like this: DecimalPart(3.4231) = 0.4231 I will be using ...
1
vote
2answers
33 views

Interpretation of definitions and logical implication in Calculus - e.g. monotonic strictly increasing function

I read definitions in Calculus books that often confuse me from a logical perspective. For example, the definition of a monotonic function, e.g. a strictly increasing function, is defined as follows. ...
1
vote
2answers
43 views

Bijection function from $(0,1]$ to $[1,\infty)$

Someone has an idea to Bijection function from $(0,1]$ to $[1,\infty)$ ? I have thought about $f(x) = \frac{1}{x}$ Is this correct ? Thanks.
3
votes
1answer
42 views

How to minimize a distance (more than one problem)

Minimizing a Distance When we seek a minimum or maximum value of a function, it is sometimes easier to work with a simpler function instead. (a) Suppose $g(x) = \sqrt{f(x)}$ where $f(x) \geq 0$ ...
3
votes
2answers
92 views

Why can't the inverse of $F(x)= x+\sin(x)$ have a formula algebraically?

I'm only curious why the inverse of $f(x)$ can not be determined algebraically. Is it because the inverse of $\sin(x)$ cannot be converted into a formula?
7
votes
2answers
46 views

Is there a name for the operation $f^{-1}(f(x) \oplus f(y))$?

This question is inspired by and/or a generalization of this question about the "reciprocal addition" operation. Consider the following: One is tempted to say multiplication is simply "addition ...
3
votes
3answers
213 views

Why does the name “epimorphism” refer to a surjective homorphism?

The wikipedia page talks about epimorphisms with category theory in mind, but I have no experience with this and ask this question from a group theory point of view (answers from any point of view are ...
0
votes
3answers
44 views

Write $f(x) = x \cdot |x|$ as a piecewise function

$$f(x) = x\cdot|x|$$ I was wondering how this function should look if I expanded it to have the format of a piecewise defined function? I know how to write a piecewise defined function, but the ...
0
votes
2answers
23 views

Suppose that g is the inverse function of a differentiable function f and G(x) =$\frac{-4}{g^2(x)}$ …

Problem : Suppose that g is the inverse function of a differentiable function f and G(x) =$\frac{-4}{g^2(x)}$ If f(5) =3 and $f'(5) =\frac{1}{125}$ then find $G'(3)$ My approach : f(5)=5 ...
2
votes
1answer
45 views

Mysterious functions

I originally asked the following question in stackoverflow, but the question is closed because some members meant that the question is about math(see the following thread) So I will give a try here: ...
1
vote
2answers
21 views

Differential equations with Euler's method

A differential equation y' + 2y = 2 - e^(-4*t) With starting point y(0) = 1 and increment ...
0
votes
1answer
23 views

Analytic and smooth functions

In my work, I first make an assumption: Assume the function $f(x)$ is an analytic function of $x$. Based on this assumption, I expand $f$ as Taylor series $$ f(x)=f_0+f_1x+f_2x^2+f_3x^3+\dots $$ ...
0
votes
1answer
56 views

Soft Question: Do most mathematicians agree that the function is “the most important concept in all of mathematics”?

Spivak (Calculus, 3e, p. 39) writes: Undoubtedly the most important concept in all of mathematics is that of a function---in almost every branch of modern mathematics functions turn out to be ...
0
votes
0answers
7 views

Graph and Stitch Piecewise Function

I am trying to develop a model to describe the rate of something increasing. It increases 88983 for 1.26s, then stops for 2.3s. It repeats this cycle indefinitely. The best I could come up with is: ...
-1
votes
3answers
48 views

Can you help me prove that this function increases? [closed]

Can you help me prove that $f(x) = x/\ln (\ln x)$ is increasing on $(e^2, +\infty)$?
0
votes
1answer
26 views

solve system inequalities derived from a function

I have this system of inequalities $$ \begin{cases} y^2-3 \geq 0\\ 16y^4-96y^2 \geq 0 \end{cases} $$ the solution for the first inequality is $y\leq -\sqrt{3}$ or $y\geq \sqrt{3}$ and the solution ...
4
votes
2answers
53 views

Functions and Derivatives

Generaly curious: Let there be a set of functions: Will the sum of the derivatives of the functions be equal to the derivative of the sums?
1
vote
3answers
95 views

Find the range of $y = \sqrt{x} + \sqrt{3 -x}$

I have the function $y = \sqrt{x} + \sqrt{3 -x}$. The range in wolfram is $y \in\mathbb R: \sqrt{3} \leq y \leq \sqrt{6}$ (solution after correction of @mathlove) $\sqrt{x} + \sqrt{3 -x} = y$ $$ ...
1
vote
2answers
23 views

Are the “weights” inside a neural network actually “terms” for a polynomial?

This just hit me today. I am not too experienced with math or neural networks, but I am trying to find out about them in my own way so I can some day understand them well. So I was thinking about how ...
1
vote
5answers
34 views

Give an example of a function from A to B that is not one-to-one. Explain why it is not one-to-one

A= {a,b,c,d} B= {1,2,3,4,5} Currently studying for a final. I know that a one-to-one function cannot map to 2 elements. There are more elements in B than in A. I don't know how to give a specific ...
2
votes
0answers
37 views

Why does the definition of a bounded function, and its negation, seem to prove contradictions?

If I want to show that $f(x) = \dfrac{1}{x}$ is not bounded above on $(0,1]$, the way I have learnt to proceed is to assume that $f$ is bounded above, so by definition there is a number $M > 0$ ...
0
votes
1answer
9 views

Find a Weight Function with specific characteristics

I need to build a weight function and I want to understand how you would do that. The reasoning you would use to define it. My function has to be something like: $f(\alpha)$ which is: $0$ if ...
-1
votes
1answer
30 views

area and volume [closed]

The total length of all 12 sides of a rectangular box is 60. (i) Write the possible values of the volume of the box. Your answer should be an interval. Now suppose in addition that the surface area of ...
1
vote
0answers
27 views

Function that satisfies the given (x,y) values

I am trying to come up with a function that (approximately) satisfies these (x,y) values. (2, 2), (3, 3), (4, 4), (5, 5), (6, 6), (7, 7), (8, 6), (9, 5), (10, 4), (11, 3), (12, 2), (13, 1), (14, 2), ...
6
votes
2answers
72 views

Conjecture function $g(x)$ is even function?

Let $f,g:R\to R\setminus\{0\}$ and $\forall x,y\in R$,such $$\color{crimson}{f(x-y)=f(x)g(y)-f(y)g(x)}$$ I have prove the function $\color{crimson}f$ odd function. because let $y=0$ we have ...
0
votes
3answers
30 views

Continuous for each variables does not implies continuous

Prove or disprove the following statement: Statement. Continuous for each variables, when other variables are fixed, implies continuous? More clearly, prove or disprove the following problem: Let ...
2
votes
1answer
53 views

How do we know which terms are of higher order?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. [...] At this point, we have shown ...
0
votes
1answer
35 views

How to Integrate by Parts when One Function Is a Polynomial?

So I have the following integration: ∫(x -2)(cosx)dx between the intervals (-a,a). we know that, The integral of a even function over the range (-a,a) can be rewritten as twice of the intergral ...
2
votes
1answer
37 views

When is a balance assumption consistent?

From Asymptotic analysis and perturbation theory by Paulsen: Find the behavior of the function defined implicitly by $$x^2+xy-y^3=0$$ as $x\to\infty$. [...] The final case to try is to ...
0
votes
0answers
17 views

Is there a word for two functions that can be defined using each other?

Let's say I have two functions f and g. For example: f(x) = x + 1 g(x) = x - 1 And I can use function g to define function f just be somehow manipulating the ...
0
votes
1answer
49 views

How to prove that $x\csc x <\pi/3$

If $x \in (0,\frac \pi6)$, then using calculus prove that $x\csc x<\frac \pi3$ My attempt: let $f(x)=\csc x$$$\implies f'(x)=-\frac{\cos{x}}{\sin^2 x}$$ which is less than $0$ for all $x\in ...
1
vote
1answer
25 views

Preimage of $[0,1]$ under $f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \frac{-27(1+\frac{1}{x^{3}-3})^{2}}{x^{3}-3}$

I want to find $$ f^{-1}([0,1]) $$ where $$f:\mathbb{C}\rightarrow\mathbb{C}, z\mapsto \frac{-27(1+\frac{1}{x^{3}-3})^{2}}{x^{3}-3}.$$ I have to do this in order to find a dessins d'enfant associated ...
0
votes
1answer
28 views

Find the function the range of the $\frac{5+2\sqrt{3+2x}-x}{\sqrt{x+1}+\sqrt{3-x}}$

Find the function range $$f(x)=\dfrac{5+2\sqrt{3+2x}-x}{\sqrt{x+1}+\sqrt{3-x}}$$ since $$\begin{cases} 3+2x\ge 0\\ x+1\ge 0\\ 3-x\ge 0 \end{cases}$$ then the function domian is $-1\le x\le 3$ and ...
-2
votes
2answers
31 views

Area of polygon inscribed in a circle [closed]

Let $A_n =$ the area of a regular $n$-sided polygon inscribed in a circle of radius $1$ (i.e., vertices of this regular $n$-sided polygon lie on a circle of radius $1$). ($i$) Find $A_{12}$. ...