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

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3
votes
3answers
42 views

Find the Range of the function $f(x) = |x-6|+x^2-1$

find the Range of $f(x) = |x-6|+x^2-1$ $$ f(x) = |x-6|+x^2-1 =\left\{ \begin{array}{c} x^2+x-7,& x>0 .....(b) \\ 5,& x=0 .....(a) \\ x^2-x+5,& x<0 ......(c) \end{array} ...
0
votes
1answer
46 views

Maximum Value of function f

How can I find the maximum value of the function \begin{equation*} f(x,y) = x^2 - y^2 + 2xy - 2x - 2y + 1 \end{equation*} where $x^2+y^2 \leq 2x$?
1
vote
3answers
36 views

Trouble understanding One-One and Onto function.

So I have a question like this: Let $g$ be a function $g : \mathbb{Z} → \mathbb{Z} \times \mathbb{Z}$ such that $g(n) = (2n, n + 3)$. And I want to find if this is onto and one-one. But I'm ...
0
votes
4answers
48 views

How to find : Range of $cos(cosx) $

How to find : Range of $\cos(\cos x) $ My approach : Since $-1 \leq \cos x \leq 1$ we get $\cos(-1) \leq \cos(\cos x) \leq \cos(1)$ Is it correct? please suggest. Can we use the above method ...
2
votes
2answers
35 views

Continuity and differentiability of the function $x|x|$

Let $f:\mathbb R \to \mathbb R$ defined by $f(x) = x|x|$, Is the function continous at all points? If it is, then is it differentiable at all points? Yes, the function is continuous everywhere but ...
0
votes
1answer
16 views

Local Extremes and Differentiable Functions

A local extreme value is found in the interior domain of a differentiable function. A claim is made that the curve must have a positive slope on one side of the extreme and a negative slope on the ...
0
votes
0answers
61 views

I need a function with Sine like movements that starts from 0 (on Y axis) and goes to 1 (on Y Axis) while only using Multiplication [closed]

Can anyone writes a function that produces the same Y s that you see below between $x = 0$ and $x = 5$ using multiplication and not addition ? (It doesn't need to be a prefect match! something similar ...
5
votes
3answers
32 views

Possible textbook redundancy concerning invertible mappings

In my textbook (Modern Algebra by John Durbin, 6th Ed), there is the following theorem: Let $S$ denote any nonempty set. (a) Composition is an associative operation on $M(S)$, with identity ...
0
votes
3answers
141 views

Endomorphic Function Definition

I need to confirm my thinking on endomorphic functions. Since an endomorphism is just a surjective morphism on an object to itself in a category, can I alter the usual definition of a surjective ...
0
votes
1answer
24 views

Is this a proof that recursive definition of functions indeed defines a function?

Someone asked me how you prove that defining a function recursively actually defines a function, and then I tried to rigorously prove it. Is it right? Let $\mathbb{N}=\{0,1,2,\dots\}$. For any ...
4
votes
1answer
60 views

Curious formula for minimum?

A few years ago I derived the following formula which I just came across in my notes: $$\min(x,y)=\log\left(\frac{e^x+e^y}{1+e^{|x-y|}}\right)=y+\log\left(\frac{1+e^{x-y}}{1+e^{|x-y|}}\right).$$ Has ...
4
votes
1answer
41 views

Simple Derivation of Functional Equation Question (L'Hospital's Rule)

First, the question is: $f$ is a differentiable function and $f : R \rightarrow R$ $xf(x)-yf(y)=(x-y)f(x+y)$ $f'(2x)=?$ My approach for problem is using L'Hospital's rule: $$ ...
3
votes
1answer
68 views

Solving an equation including $e^{-x}$ with the Lambert W function

Given two functions of $x$, namely $f(x)$ and $g(x)$, where $$f(x)=x^2-4x+8$$$$g(x)=3xe^{-x}$$ the shortest distance between the graphs of the functions is sought. I begin by defining a function ...
0
votes
0answers
78 views

Centripetal Catmull–Rom spline

What is "t" in this short and simple example below? There are 4 points Pn[xn,yn] in 2D space: A[1,6] B[3,1] ...
0
votes
1answer
90 views

Inverse of $x^x$ [duplicate]

Since $x^x$ grows very fast, its inverse should accordingly grow very slow, possibly slower than $\ln(\ln(x))$. I am troubled with finding such an inverse: I only get to the point: $\ln(x)x=\ln(y)$ ...
3
votes
3answers
36 views

$\sqrt{y}+\sqrt{x}=\sqrt{A}$ … prove that x-intercept + y-intercept of any tangent = constant [closed]

This is equation of a curve $\sqrt{y}+\sqrt{x}=\sqrt{A}$ $A$ is constant $T$ is a tangent of the curve from any point on it $B$ is y-intercept of $T$ $C$ is x-intercept of $T$ ...
0
votes
1answer
31 views

General expression that represents a combined period of 2 sine functions

How to find the general expression that represents the combined period of $y=5\sin(\pi x/6)$ and $y=3\sin(\pi x/4)$? what are the limitations of this model?
-1
votes
1answer
20 views

Need help with a hyperbola/parabola equation solving for an assignment

Hi so i was given my Math C assignment today and the moment i looked at question 1 i knew i had no idea what to do. This is the graph i was given (http://imgur.com/nRXOlJy). I was asked to provide an ...
0
votes
1answer
24 views

How many license plates could begin with A and end with Zero

Suppose that in a certain state , all automobile license plates have four letters followed by three digits. A) How many license plates could begin with A and end with Zero B) How many plates are ...
4
votes
0answers
41 views

Finding period of a function? Am I doing something wrong?

I need to find the period of the function $$\large{\frac{\sin (\sin {nx})}{\tan{\frac{x}{n}}}}$$. According to me, the period of $\sin (\sin (nx))$ should be $\large{\frac{2\pi}{n}}$ and the period ...
0
votes
2answers
28 views

Injective implies invertible? Injective and well-defined implies bijective?

I have two questions regarding functions regarding linear maps: (Let $X$ and $Y$ be to Banach spaces) If $T:X\rightarrow Y$ is injective, then $T^{-1}$ exists, right? If $T:X\rightarrow Y$ is ...
0
votes
0answers
31 views

Airy function integral

I have seen this result stated in a paper $$\frac{\textrm{Ai}(\xi_{0})}{\int_{\xi_{0}}^{\infty}\textrm{Ai}(\xi)\,\textrm{d}\xi}=1.001\textrm{i}^{1/3},$$ where $\xi_{0}=-2.298\textrm{i}^{1/3}$. I ...
1
vote
2answers
24 views

What is the name of the Boolean function whose output is always one?

For example: f = a.b.c.d + !a.!b.!c.!d + a.!d + !a.b.!c + !b.d + b.c.d + a.b.!c.d + !a.c.!d = 1 ! is logical NOT, . is logiacal AND and + is logical OR. The ...
0
votes
0answers
10 views

domain and range of a function

I'm supposed to get the domain and range of this function of two variables f(x,y)= sin([[x + y]]pi/2) ([x+y] is a step function). I don't have anyone to teach me this because I don't go to school and ...
1
vote
3answers
14 views

Domain and range

I just need some clarifications. I'm given a function of two variables $f(x,y)=2-x^2-y^2+2x-4y$ and I'm asked to find the domain and range of it. Now I know that the domain of this is all real ...
0
votes
1answer
66 views

prove that the equation has just one root.

prove that the equation $$2^x + 3^x + 4^x - 5^x =0$$ has just one root. ATTEMPT: Write $2^x + 3^x + 4^x = 5^x$. By sketching the graphs it is confirmed that they will intersect at somewhere ...
0
votes
1answer
18 views

Define domain of the function

Is it right to determine the domain of the function $f(x) = (x - 2)(8 - x)$, $x\in\Bbb R$ as $D(f) = \{y\in\Bbb R \mid y \geq -9\}$? $f(x) = x² - 10x + 16$ $\Delta = 36$ $\therefore$ $x_1 = 8$, ...
0
votes
3answers
85 views

How can I prove that $f(| a|) = -|a|$? [closed]

Set the function $f(x) = |x| - 2x,~x\in\mathbb{R}$. Prove that $f(|a|) = -|a|$. How can I do this? I have no idea.
1
vote
2answers
38 views

Little confused about the constraint of Injective Functions and Surjective.

From my understanding, A Function is called to be Injective, if different elements of the first set are mapped to different elements of the second set. Let set A = {a,b,c} and set B = {1,2,3} Are ...
1
vote
3answers
43 views

Prove that $\Gamma\left(-a\right)=\left[\Gamma\left(a\right)\right]^{-1}$ for $\Gamma:\mathbb{Z}\rightarrow \mathcal{B}\left(A,A\right)$

I am working through various problems in Bloch's Proofs and Fundamentals and I'm stuck on this problem (in need of hints): Let $A$ be a set. A $\mathbb{Z}$-action on $A$ is a function ...
1
vote
1answer
35 views

Confused on the argument of this function?

So say I wish to go from $$12\sin (t)+4\cos(t)$$ to the form $$A\cos (t+k)$$ by using the double angle formula I can get that $$\cos(k)=4$$ and $$\sin(k)=-12$$ and so we can find ...
0
votes
0answers
12 views

How to determine the period of the following functions?

How would the following make a difference to the period of a function? $$ \cos(t)~~~~ (1)$$ $$ \cos(\omega t)~~~~ (2)$$ $$ \cos(\omega t + \phi)~~~~ (3)$$ Would this be right, $(1)$ has period ...
1
vote
1answer
32 views

What is the meaning of expressions of the type $f(\cdot)$ (function (dot))?

Simple question, fully expressed in the Title line. Is the dot within the parenthesis intended to mean, "any possible function"?
5
votes
3answers
75 views

Prove that $f: [a,b] \rightarrow \mathbb{R}$ is strictly monotone

I'm a 1st year mathematics student, and in my analysis class I'm having trouble with proving the following: Let $a < b$ be real numbers, and let $f: [a,b] \rightarrow \mathbb{R}$ be a function ...
0
votes
0answers
12 views

Name of this function

Is there a name for the following function? $f(x,y) = \begin{cases} y+1, & \text{if $x=0$} \\ f(x-1,1), & \text{if $y=0$} \\ f(x-1, f(x-1,y-1)), & \text{otherwise} \end{cases}$
8
votes
8answers
237 views

An example of a mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that $\eta(x)=n$ has infinitely many solutions for each $n\in\mathbb{N}$

Suppose I have the mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\eta(x)=n$ has infinitely many solutions. I saw this question which is basically the ...
0
votes
1answer
15 views

How does one show quasiconcavity?

Is there a general method, or does one have to get creative in some way? For example, let $$f(x,y,z) = 3\sqrt{x + y + 2z}$$ I've tried looking at the definition, but I am not sure what to do with ...
1
vote
1answer
22 views

First order approximation of $F(x)=\int_0^x f(t) dt$ in the neighbourhood of $\infty$

Let $f(x)$ continuous on the real line. Then the first order approximation of $$F(x)=\int_0^x f(t) dt$$ in the neighbourhood of $0$ is: $$F(x)=\int_0^x f(t) dt\sim 0 + x f(0), \ \ \ (x\rightarrow 0)$$ ...
1
vote
1answer
15 views

How to determine a function from a sequence of consecutively composed functions?

Let $ f(x) = x+1 $ and $g(x) = 2x$ Prove $f^2g = gf $ and determine $f^igf^jgf^k(x)$ explicitly as a function of x and in terms of i,j,k. I got through the proof but I don't understand what the ...
1
vote
3answers
46 views

How to prove $g$ is discontinuous at $x=2$ using definition of limit?

Define $g: \mathbb{R} \to \mathbb{R}$ by $$g(x) =\begin{cases} 5x-15 & \text{if } x \text{ is rational}, \\\\ x^3-17 & \text{if } x \text{ is irrational}. \end{cases}$$ Prove that ...
0
votes
1answer
29 views

How to prove that a $\phi \in C^{\infty}(\mathbb{R})$.

I would like to prove that the function, defined as: \begin{equation} \phi(x)=\begin{cases} e^{-1/x}, & x>0 \\ 0 , & x \leq 0\end{cases} \end{equation} is a $C^{\infty}(\mathbb{R})$. So ...
1
vote
3answers
52 views

How to find the remainder of polynomial division?

Im trying to solve this problem but I do not understand what the question is asking: Let $n\ge 2$ be an integer and $ p_n(x) $ be the polynomial: $$ p_n(x) = (x-1)+(x-2)+\cdots+(x-n) $$ What is the ...
1
vote
1answer
31 views

How do I prove this inequality using the Mean Value Theorem?

Let $f: \mathbb{R} \to \mathbb{R}$ be a function (that's not necessarily continuous, bounded, or any other property) and let $h:\mathbb{R} \to \mathbb{R}$ be a differentiable function such that for ...
7
votes
2answers
76 views

Roots of a polynomial whose coefficients are ratios of binomial coefficients

Prove that $\left\{\cot^2\left(\dfrac{k\pi}{2n+1}\right)\right\}_{k=1}^{n}$ are the roots of the equation $$x^n-\dfrac{\dbinom{2n+1}{3}}{\dbinom{2n+1}{1}}x^{n-1} + ...
0
votes
2answers
40 views

How to find the positive and negative roots of a function?

Iam trying to solve the following question: Find all numbers $a$, such that the equation $x^2-ax-a = 0$ has one positive root and one negative root. I've tried it already but I cannot seem to ...
4
votes
4answers
73 views

If $f'(c) \neq \frac{f(b)-f(a)}{b-a}$, then find number of such $c$.

Let $f(x)=x^3+3x+2$ and $x=c$ is a point such that $$f'(c) \neq \frac{f(b)-f(a)}{b-a}$$ for any two values of $a$ and $b$, where $a,b$ and $c \in \mathbb R$. Then find the number of ...
0
votes
4answers
35 views

How to find maximum value of trig function?

How to find maximum value of this: $$y = 5\sin x - 12\cos x$$ And I am more intrested in solving process, rather than answer. I know the answer. I am familiar with derivatives, not so good, but as I ...
0
votes
1answer
19 views

is this the way to evaluate these functions? if not, how can I evaluate?

Given that $$ f(x)=1-x, g(x)=x^2-3, h(x)=\frac2x $$ 1) Find $$f\,o\,g$$ My answer 1) $$f\,o\,g(x)=f(g(x))=f(x^2)=3x^2-1$$ 2) Find $$f\,o\,h$$ My answer 2) $$f\,o\,h = f(h(x))= 2/1-x-3 $$ 3) ...
0
votes
4answers
66 views

Limit with the sinus squared function

In my assignment I have to determine if the following limit is true or false: $$\lim_ {x \to 0} \frac {\sin^2(2x)}{\sin(2x^2)}=1$$ I think that it's false. Here is my calculation, and I wanted to ...
4
votes
1answer
44 views

Trigonometric root of a polynomial

If $4\cos^2 \left(\dfrac{k\pi}{j}\right)$ is the greatest root of the equation $$x^3-7x^2+14x-7=0$$ where $\gcd(k,j)=1$ Evaluate $k+j$ I tried factorizing the equation but it wasn't ...