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

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0
votes
1answer
21 views

How to find the domain of this trig function?

f(x)=sqrt(tan(2x+π)) Allright, so i know you cannot have a number less than zero under the square root sign and that tangent cannot equal π/2+nπ. So should i try to find the domain of the tan ...
0
votes
1answer
49 views

When the imaginary part of a function is zero?

Let $z_k=x_k+ i y_k, x_i,y_i \in \mathbb{R}$ are the complex variables. Consider a polynomial of $z_k$ and its conjugates $f(z_1,\ldots,z_n, \bar{z}_1, \ldots,\bar{z}_n).$ Question:Is there any ...
3
votes
2answers
33 views

Another question about $x_0$ in the Taylor series

When we talk about Taylor series, we say it's around point $x_0$. It's in the Taylor series formula: $$f(x) = f(x_0) + (x-x_0)f'(x_0) + \frac{(x-x_0)^2f''(x_0)}{2} + \frac{(x-x_0)^3f'''(x_0)}{6} + + ...
1
vote
1answer
31 views

Find the max and the min of the function

I have this function: $$g(t)=t^2+\cos(2t)-\cos(t)$$ $$0\le t\le2\pi$$ I made the derivative:$$g(t)'=2t-2\sin(2t)+\sin(t)$$ And except for the obvious solution $t=0$ I'm not able to find the other.. ...
0
votes
2answers
37 views

To find inverse of function [on hold]

Given $ f(x) = \begin{cases} 2x, & \text{if $x\in[0,1]$} \\ 8 - 2x, & \text{if $x\in [2,3)$} \end{cases} $ Then how to find inverse of f ?
0
votes
2answers
52 views

Find the range of $ x-\sqrt{4-x^2}$

$Y=x-\sqrt{4-x^2}$. How to find these types of functions' range? I just know that the answer is $R=\{y\in\mathbb{R}\mid-2\sqrt{2}\leq y\leq 2\}$, but I have no idea how to find it step by step.
0
votes
1answer
25 views

Can the domain and co-domain be the same set? Is this a function?

Let $A$ denote the set of all real numbers. Let $B$ denote the same set as $A$. Let $f$ be the function that, to each number in $A$ assigns the cube of the number. Is $f$ a function?
1
vote
2answers
35 views

Functions - finding the domain

Question: Consider the function: $$f(x) = \log(2x + 1) - \log(x - 3)$$ What will be the domain of this function? I used two approaches to solve this question. Both approaches got me different ...
0
votes
1answer
25 views

Smooth saturation function

I need a function similar to $$Saturation(x)=min(max(x, -1), 1)$$ except for I need it to be smooth with no jump in its derivatives. It seems $arctan$ is not a good candidate since I need it to keep ...
0
votes
1answer
21 views

Total number of distinct solution produced by polynomial

I have a function $F(x,y) = ax + by$ where $x,y$ belongs to range $[1..10^{10}]$ and $a$ and $b$ are constants, all are integers. How many distinct values can be produced by this function, please give ...
0
votes
1answer
35 views

If the cos of 27 is 0.89, how much is the csc of 27

Hey guys for my trig class we're viewing trigonometric functions and their properties. So far I have understood but I came across this problem and can't seem to solve it: Given the approximation cos ...
1
vote
1answer
20 views

Function that maps a rational number to its numerator and denominator

Question: Is there a simple way to represent a function $f:\mathbb Q\to \mathbb Z^2$ that maps a rational number in lowest terms $r=\frac ab$ to the ordered pair of its numerator and denominator ...
0
votes
1answer
24 views
+50

Understanding relation between vector valued function and function objective in an multi objective optimization problem

I try to understand the relation between "vector-valued function" and "function objective" as used in optimization problem. I understand that objective function in a multi-objective problem can be ...
0
votes
1answer
30 views

What is the highest order of derivative of this function $f(x) = x^5\sin(\frac{1}{x}) $ at $x=0$?

The function is defined as $f(x) = x^5\sin(\frac{1}{x}) \quad \text{for} \quad x\neq 0 \quad $ and $f(x) = 0$ for $x=0$. I can't tell by just looking at the plot. I think there might be a theorem I ...
2
votes
0answers
19 views

A question about a notation used in the Folland Real Analysis

This is the exercise 11 in the Folland Real Analysis. Could anyone tell me what it means by f(x,・) and f(・,y)? I have never seen such notations before...
6
votes
0answers
115 views
+100

Approximation of the exponential

Let $c>1,k\in\mathbb{N}$. Let's consider two approximations of the exponential function : The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$ and the second one is ...
0
votes
5answers
66 views

Find the limit of the function [closed]

Let we have the following function $$F(x)=\frac{x^x-x}{\ln(x)-x+1}$$ Find $$\lim_{x \to 1}\frac{x^x-x}{\ln(x)-x+1}$$
20
votes
5answers
3k views

Why can a circle be described by an equation but not by a function?

In high-school math functions always looked to me just like glorified equations. The only time I saw a meaningful difference was when we covered the equation of a circle and I realized that an ...
0
votes
1answer
28 views

From a point-wise to a linear piece-wise definition

Consider the following definition: $$f(x, \alpha) = \left\{\begin{array}{cl}A(x) & \alpha = 0\\ B(x) & \alpha = 0.5 \\ C(x) & \alpha=1 \end{array}\right.$$ where we always have $A(x) < ...
0
votes
0answers
25 views

Applying implicit function theorem to function with derivative

This may be a very peculiar question, or I may even be on the completely wrong track, so I apologize in advance for obvious errors. I am trying to apply the implicit function theorem in an ...
0
votes
1answer
23 views

Periodic functions proof

I need some help here. Let $f$ be a $2\pi$-periodic function, and define for an arbitrary $k\in\mathbb N$ a function $g(x) = f(kx)$. Show that $g$ is also $2\pi$-periodic. What I've done: $$ g(x) ...
0
votes
2answers
16 views

Defining domain in complex plane

I am asked to define the domain for the following given that $z=x+iy$: $a) \quad f(z) = \dfrac 1 {z^2 + 1}$ $b) \quad f(z) = \dfrac 1 {1 - |z|^2}$ How would this be different from a normal domain ...
-4
votes
1answer
32 views

Exponential Math Functions [closed]

Iodine — 131 is a radioactive substance used in nuclear medicine. Suppose a patient was given a dose of 6mL. The half-life of Iodine-131 is 8 days. Determine the amount of iodine-131 in the patient ...
3
votes
2answers
50 views

Can we give a bound on any associative function?

We say that $f:[1,\infty)^2\to[1,\infty)$ is associative if $$f(f(a,b),c)=f(a,f(b,c))$$ And symmetric if $$f(a,b)=f(b,a)$$ e.g. the arithmetic operations '+' and '$\cdot$' are associative and ...
0
votes
1answer
7 views

question about vacuous truth and function

I'm confusing about vacuous truth. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ where $f(n)=2n$. we can calculate function values if $n$ belongs to domain. but what if it does not? The value of ...
1
vote
1answer
75 views
+50

I cannot make the mental leap from a vector to a function!

In my linear algebra book, it says that a vector is linearly independent if $\vec V = c1*\vec T_1 + c2*\vec T_2$ And then it goes on to say that $y(t) = c1 * e^{-at} + c2*e^{-bt}$ is linearly ...
1
vote
0answers
9 views

Tigher bounds on concavity of log

Is there a tighter upper and lower bound on concavity of $\log(.)$ function? It is very well known that $\log(\sum_{i}p_ix_i) - \sum_{i}p_i\log(x_i) \geq 0$. But are there stronger versions of this ...
1
vote
1answer
13 views

How to interpret algebraic relationship/ next step to take to prove function is onto?

This is a problem from Discrete Mathematics and its Applications Book's definition on bijection Book's definition on onto Book's definition on one to one I am trying to do problem 23D. Here ...
-1
votes
1answer
21 views

Find the domain of this function [closed]

How would I go about finding the domain of this function? $$f(x)= \sqrt{\tan(2x+\pi)}.$$
1
vote
3answers
41 views

Solve for $y$ in $x=\sqrt{(y-1)/(y+1)}$

I always struggle with this: Express $y$ in terms of $x$ where $$x = \sqrt\frac{y-1}{y+1}$$ I know to square both sides and get $x^2 = \frac{y-1}{y+1}$ Then I'm thinking multiply both sides ...
0
votes
2answers
69 views

Proof of $\cos(y)$ and $\sin(y)$ using $e^{iy}$

I need to use that $e^{iy} = \cos y + i \sin y$ (for $y \in \mathbb{R}$) to prove that $$\cos y = \frac{e^{iy}+e^{-iy}}{2}$$ and $$\sin y = \frac{e^{iy}-e^{-iy}}{2i}$$ I'm really clueless, any ...
0
votes
0answers
26 views

How to find an injuctive function for not divisible by 7 but divisible by 5?

This is based off my other question - How to write a function to express not divisible by 3? This is a problem from Discrete Mathematics and its applications I am currently on 4B. Here is my work ...
0
votes
4answers
34 views

Surjectivity and injectivity of $\lceil n/2\rceil$

Problem: is this one-to-one, onto, or both? $$f:\mathbb Z\to\mathbb Z; n \mapsto \left\lceil \frac n2\right\rceil$$ With help I arrived at the answer is that $f$ is onto. However, I'm confused ...
0
votes
2answers
19 views

Lagrange method over two constraints

plane $x+y-z=-2$ intersects $z^2=x^2+y^2$ I need to use Lagrange multipliers to determine the point of intersection which is the closest to the origin. As far as I understand, to use Lagrange I need ...
2
votes
1answer
33 views

Domain of a composition of log functions

Domain of $$\log_3(\log_{1/3}(\log_4(\log_{1/4} x)))$$ Please guide me to solve this problem. Since it is a composition of logs, I am confused how to start.
2
votes
0answers
31 views

$\frac{\sin(nx)}{\sin(x)}=(-4)^{(n-1)/2} \prod_{1\leq j \leq (n-1)/2}(\, \sin^2(x)-\sin^2(\frac{2\pi j}{n})\,)$

In Serre's A Course in Arithmetic, it states For $n$ odd and positive integer, proof that $\frac{\sin(nx)}{\sin(x)}=(-4)^{(n-1)/2} \prod_{1\leq j \leq (n-1)/2}(\,\sin^2(x)-\sin^2(\frac{2\pi ...
-1
votes
1answer
47 views

Write$\frac{\sin(nx)}{\sin(x)}$ as polynomial in $\sin^2(x)$ [closed]

How to write $\frac{\sin (nx)}{\sin(x)}$ as a polynomial of degree $\frac{(n-1)}{2}$ in $\sin^2(x)$, where $n$ is a positive odd number.
3
votes
2answers
100 views

Solutions of the functional equation $f(2x) = \frac{f(x)+x}{2}$

How can I solve the following functional equation? $$f(2x) = \frac{f(x)+x}{2},$$ for $x \in \mathbb{R}$ with $f$ being a continuous function.
0
votes
1answer
34 views

Complex analysis basics

If I z = x + yi and w = f(z), describe the image R of D in the w-plane when $$0<x<\pi/2, 0<y<\infty;w = e^{iz}$$ Wouldn't this mean that in the w-plane the argument arg(w) = $\infty$ ...
1
vote
1answer
35 views

Prove that indicator function of integer numbers is positive semidefinite

How to prove that the function $\mathbb{1}_{\mathbb{Z}}(x)$ is positive semidefinite? I.e. to show that for any $n = 2, 3, ...$ and $x_1, ..., x_n \in \mathbb{R}$, $z_1, ..., z_n \in \mathbb{C}$ ...
0
votes
1answer
53 views

$f(x)$ as a difference of two increasing functions

Let $f(x)$ be a continuous function. Find $g(x)$ and $h(x)$ - two increasing functions, which difference equals $f(x)$, e.g. $f(x)=g(x)-h(x)$. Examples: $\arctan(x^3-9x)$ $\frac{1}{1+(\sin x)^2}$ ...
0
votes
1answer
25 views

Graphing Quadratic Function Describing a Parabolic Arch [closed]

An architect decides to use a parabolic arch for the main entrance of a science museum. In one of his plans, the top edge of the arch is described by the graph of $y=-\frac{1}{4}x^2+\frac{5}{2}x+15$. ...
0
votes
0answers
20 views

Algebraically evaluate functions and express answer as a single fraction [closed]

$(f - g)(x)$ as a single fraction in un-factored form given $$f(x) = \frac{x-2}{2 x + 1}$$ and $$g(x) = \frac{x - 2}{2x + 1}$$
0
votes
3answers
34 views

A problem in calculus mean value theorem

Hi tried to solve this for hours, any idea how to approach this question: prove for every $x>0$ $$2x\times\arctan(x)>\ln(1+x^2)$$
0
votes
2answers
31 views

Binary relation of composite function

Suppose S is a binary relation on a set X. If S ◦ S is reflexive, Is S is reflexive? can we prove this with example too and by definition "Let U be a non-empty set and let R be a binary relation ...
2
votes
1answer
64 views

improbable sum of random variables

Let $U$ be a uniform random variable on the interval $[0,1]$. It is exceedingly unlikely that $U$ can be written as a sum $U = X + Y$ where $X$ and $Y$ are independent identically distributed random ...
2
votes
1answer
35 views

Maximum value of $f(x) = \log_{(\tan x + \cot x)}(\det A)$ for a diagonal matrix $A$

If $$A =\begin{pmatrix} d_1 & 0 & 0 & 0 \\ 0 & d_2 & 0 & 0\\ 0 & 0 & d_3 & 0\\ 0 & 0 & 0 & d_4\\ \end{pmatrix}$$ ...
0
votes
2answers
30 views

Understanding complex functions in w - and z - plane

I have a difficulty understanding the basics of complex functions. My exercise looks like this: "The $z$-plane region $D$ consists of the complex numbers $z = x + yi$ that satisfy the given ...
1
vote
1answer
12 views

To find the number of possible functions with a given property

How many $f$ are possible having this property: $f:\{1,2,....n\} \rightarrow \mathbb R$ such that $\big(\sum_{i=1}^{10}\dfrac{|f(i)|}{2^i})^2=(\sum _{i=1}^{10}|f(i)|^2)(\sum ...
0
votes
0answers
12 views

Deterministic seeded shuffle

How can I find an injective function $f$ so that mapping that function over each element of the ordered sequence $[1\cdots{n}]$, yields a deterministic shuffle (random permutation) that is "good" ...