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

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2
votes
4answers
91 views

Sufficient condition for convexity

Let f:$ [a,b] \rightarrow \mathbb{R} $ a continous function such that $ \forall (x,y) \in [a,b]^{2}, \exists t \in ]0,1[, f(tx+(1-t)y) \le tf(x) + (1-t)f(y) $ show that f is convex
0
votes
2answers
35 views

How Would I Graph This Exponential Function?

How Would I Graph This Exponential Function? $f(x) = \frac{-3}{2^{(x+2)}} - 1$, How I do it: I know that it is basically $-3\times ({\frac{ 1}{2}})^{x+2} - 1$, which is then $-3 \times 2^{-(x+2)} - ...
0
votes
0answers
15 views

Finding the value of $K$ by the replacement of a function

if $e^{f(x)}=\frac{10+x}{10-x}, x\in (-10,10)$ and $f(x)=kf\left(\frac{200x}{100+x^2} \right)$, then $k=$ (a) 0.5 (b)0.6 (c)0.7 (d) 0.8 Answer: (a) I tried by taking natural log both the ...
0
votes
0answers
21 views

Counting the number of elements $x$ between $p$ and $p^2$ where lpf$(x(x+2))=7$

Let $p > 7$ be any prime. Let $f_7(p)$ be a function that counts the number of elements $x$ where $p < x < p^2$ and lpf$(x(x+2))=7$ where lpf is the least prime factor. It has been ...
0
votes
1answer
20 views

Does the function have horizontal or vertical asymptotes?

So I'm analyzing some functions here and I need to determine whether or not they have horizontal or vertical asymptotes. The equations are: $f(x)=260$ $g(x)=1+24(0.9)^x$ $h(x)=f(x)/g(x)$ Now ...
2
votes
3answers
27 views

Base of the $\mathbb{R}$ vector space that contains all real functions: $f(x) \not= 0$ for finitely many x $\in\mathbb{R}$

I did already prove that this is a vector space. It is easily shown that addition and scalar multiplication with functions that hold the above property again yields a function with $f(x) \not= 0$ for ...
10
votes
1answer
186 views

Solving a special Quartic Equation.

Solve for $x$ $$(x^2-4)(x^2-2x)=2$$ I have tried the Rational Root Theorem and found that there are no rational roots. Further, the polynomial $p(x)=(x^2-4)(x^2-2x)-2$ is irreducible since ...
1
vote
2answers
12 views

Horizontal and Vertical Asymptotes of functions

So I'm completing a chart analyzing the different properties of three different functions: $f(x)=\log(x^2+6x+9), g(x)=\sqrt{x^2 -1}$ (sorry not sure how to do square roots on here), $h(x)=f(x)(g(x))$ ...
7
votes
2answers
136 views

Polynomial Functional Equation.

Let $f(x)$ be a one-one, polynomial function such that $f(x)f(y)+2=f(x)+f(y)+f(xy) \ \forall \ x,y \in \mathbb R - \{0\}$, $f(1) \neq 1$, $f'(1)=3$. Find $f(x)$. I tried to find the degree of ...
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
2answers
30 views

What's the minimum value of the following function?

So, I need to figure out the minimum value of this function: \begin{equation*} y=x^2-2(m+1)x+2m(m+2). \end{equation*} I tried with the y-coordinate of the parabola's tip, but all I get is the ...
1
vote
1answer
67 views

Continuity of $f(z) = Log z$ , for $z$ complex, non-real $-\ln|z|$, for $z$ real

At what points in the complex plane is this function continuous (If there is any)? Would it be correct to conclude that $f$ is then continuous for all $z$ in the complex plane less the real ...
0
votes
2answers
29 views

How to show $x,y,z \in A$ - Functions, Combinatorics

If $A \subseteq \{1,2,3,4,5,6\}$, how to show that for every $A$ there are $x,y,z \in \{1,2,3,4,5,6\}$, where $x,y,z$ can also be the same or at least not different from each other, and the following ...
7
votes
2answers
75 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
1answer
22 views

What is this octagon constant and how do I calculate it for other 8*N-gons?

I'm drawing a circle with triangles in OpenGL and I am no good at maths. I've tried a couple of ways, one including the simple ...
0
votes
1answer
25 views

Show that the function $g$ is bijective

Let $~f\colon X \!\to\! \{i \in \mathbb{N}:\! 1 \leq i \leq n \}$ be a bijective function and $~x$ be an element of $~X$. Now define the function $g\colon X-\{x\}\!\to\!\{i \in \mathbb{N}:\! 1 \leq i ...
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
0answers
60 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 ...
0
votes
1answer
44 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 $x^2+y^2 \leq 2x$?
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 ...
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
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 ...
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 ...
4
votes
1answer
59 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 ...
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 ...
4
votes
1answer
40 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: $$ ...
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 ...
8
votes
8answers
235 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
0answers
46 views

Looking for a function which can serve as an upper bound to a count of the the pairs (x)(x+2) that have a given least prime factor?

Let $p \ge 7$ be a prime. Let $z > p$ also be a prime. Let $f_p(z)$ be the number of elements $x$ such that $z \le x < z^2$ and the least prime factor of $x(x+2) = p$ I am trying to find ...
0
votes
1answer
89 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 $T$ ...
4
votes
2answers
170 views

What does it mean to have a lone plus sign in the exponent/superscript (Modified Weiszfeld algorithm)

I was skimming through Vardi & Zhang's paper on "A modified Weiszfeld algorithm for the Fermat-Weber location problem". It is available at http://www.stat.rutgers.edu/home/cunhui/papers/43.pdf ...
0
votes
1answer
30 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?
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
40 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
2answers
132 views

Symbolic rule help

This is a basic algebra question, I could use a little assistant with I just want guidance with symbolic rule. A photocopier was bought for 4000 dollars and depreciates at a rate of $500 per year. ...
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
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.
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 ...
1
vote
3answers
151 views

Showing a mapping has infinitely many solutions

Problem: Define a mapping $\gamma\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\gamma(x)=n$ has infinitely many solutions. Book solution: Let $p_n$ denote the $n$th ...
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 $D(f) = \{y\in\Bbb R \mid y \geq -9\}$? $f(x) = x² - 10x + 16$ $\Delta = 36$ $\therefore$ $x_1 = 8$, ...
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 ...