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

learn more… | top users | synonyms (1)

5
votes
2answers
425 views

A polynomial of degree 3 that has three real zeros, only one of which is rational.

Find a polynomial of degree 3 that has three real zeros, only one of which is rational. My answer: $(x - \sqrt{2})(x - 3)(x - \pi)$. Is this correct? It does have two irrational zeros, but I'm not ...
-1
votes
0answers
25 views

Differentiable function f(x)

Let $f(x)$ is a differentiable function satisfying $f'(x) + 100 f(x) ≤ 1 $ Then $f(x) -1/k$ is a non increasing function of $x$ , then we have to find the value of $k $ I tried , but at last ...
1
vote
0answers
48 views

A curious trigonometric equality

Let's consider the following expression: $(1)\cos(15\sqrt{2}^\circ) = \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}+\frac{i}{2}} +  \frac{1}{2}\sqrt[\sqrt{2}]{\frac{\sqrt3}{2}-\frac{i}{2}}$ The left ...
3
votes
4answers
92 views

Solving a functional equation ( $ f(x-y) = f(x)/f(y)$ )

Consider the functional equation $$f(x-y)=f(x)/f(y)$$ If $f'(0)= p$ and $f'(5)=q$, then what is the value of $f'(-5)$ ? My attempt. Using the equation written above I was able to determine the ...
0
votes
0answers
9 views

Find the maximum value of $f(x,y,z)$ on the interval $x_0<x<g^x(p)$, $y_0<y<g^y(p)$, $0<z<g^z(p)$, $p=p(x,y,z)$

First of all, sorry if I am misusing terms or any tags in the post; I am a bit out of my depths here so I'm just trying to explain things in layman's terms. Now, here's the problem: I am working on ...
-1
votes
3answers
38 views

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$. [closed]

Given that an expression $2x^3+px^2-8x+q$ is exactly divisible by $2x^2-7x+6$, determine the value of $p$ and $q$.
0
votes
0answers
14 views

Checking of uniformly continuity of the following functions

Which of the following 4 functions are uniformly continuous? and which are not? I want to know the process/explanation of the solutions.
0
votes
1answer
33 views

Hi I was wondering if there is any algebraic way to find the zeroes of a cos/sin formula without using the unit circle? [closed]

I understand how to find the zeroes using the unit circle or just graphing it for any matter for my equation. I was just wondering if there is a formula or an algebraic way I could find them. Here'...
1
vote
2answers
34 views

Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto

Let $f:A \to B$ and $g:B \to C$ be maps. Prove that if $g \circ f$ is onto and $g$ is one-to-one, then $f$ is onto. Attempt: If $g \circ f$ is onto, then for all $y \in A$, $\exists x$ such that $...
4
votes
1answer
20 views

Constructing bijection from set of equivalence classes to another set

Suppose $f:A \to B$ is surjective. Define a relation on $A$ by setting $x\sim y$ if $f(x) = f(y)$. It is clear that $\sim$ is an equivalence relation on $A$. Let $\mathcal{E}$ be the set of ...
1
vote
2answers
42 views

If $ Q(x)= x^2-5x+1 $ , find $ \frac {Q(5+h)-Q(5)}{h} $

If $ Q(x)= x^2-5x+1 $ , find $ \dfrac {Q(5+h)-Q(5)}{h} $ can someone show the steps to reach the answer of $h+5$? I got it down to $\frac{h^2+5h-2}{h}$
2
votes
2answers
21 views

Function that returns 1 when a non whole number, 0 when whole number

The title in this case should be self explanatory. When $x$ has a fractional part greater than $0, y$ should be equal to $1$, and when $x$ is a whole number, $y$ is equal to $0$. Anything that gives ...
2
votes
2answers
12 views

Checking injectivity of a certain function from a union of a family indexed by $K$ to $K$

Let $A = \{ A_k | k \in K \}$ be a family of sets indexed by $K$. By Zermelo's theorem, $K$ can be well-ordered. Now, let $\leq$ be a well-order on $K$. Define $j: \bigcup\limits_{k \in K} A_k \to ...
1
vote
2answers
42 views

Check if $f(x)=2 [x]+\cos x$ is many-one and into or not?

If $f(x)=2 [x]+\cos x$ Then $f:R \to R$ is: $(A)$ One-One and onto $(B)$ One-One and into $(C)$ Many-One and into $(D)$ Many-One and onto $[ .]$ represent floor function (also known as greatest ...
2
votes
2answers
80 views

How many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. A few related questions inside.

I am trying to calculate how many maps $A \overset{f}{\rightarrow} A$ satisfy $f \circ f = f$ with the given set $A=\{a, b, c\}$. I would like to see the explicit mappings and learn how you ...
0
votes
0answers
18 views

function nondecreasing in both variables, set of discontinuities is a nullset

Let $f\colon [0,1]^2\to\mathbb{R}$ be a function such that $g(x):=f(x,y)$ for any $y$ and $h(y):=f(x,y)$ for any $x$ are nondecreasing functions (the second variable is fixed). Prove that the set of ...
1
vote
2answers
47 views

Showing there is a constant for which an inequality holds true

I'm supposed to show that for $x>0$ and $p>0$ there is a constant $C$ such that $e^x\ge Cx^p$. The constant $C$ depends on $p$ but not on $x$. After analysing the behaviour of the graphs of ...
0
votes
3answers
57 views

If $f(x+1)= -f(x-1)$, prove that the function f is a periodic function and its period is $4$

I sure do know that I should arrive to the point where $f(x)= f(x+T)$. I've tried replacing $x+1=a$ but the problem seems to get more and more complicated.
1
vote
3answers
31 views

State the domain of $f^{-1}$

$$f(x)=\sqrt{2x+5}$$ $$x \geq -2.5$$ State the domain of $f^{-1}$ \begin{align} \ x & = \sqrt{2y+5} \\ \ \Rightarrow f^{-1}(x) & = \frac{x^2-5}{2} \\ \end{align} The Mark Scheme says that ...
1
vote
0answers
23 views

Machine Learning: are there other functions similar to the softmax?

Recall in probability and machine learning softmax is defined as: $\sigma(\mathbf{z})_j = \dfrac{e^{z_j}}{\sum_{k=1}^K e^{z_k}}$ for $j = 1, ..., K.$ where $\sigma: \mathbb{R}^k \to (0,1)$ ...
1
vote
0answers
4 views

Analytical Representation of a Sparse Matrix.

I have a sparse, rectangular array where sparse means that missing entries in the table are non-existant, not zero. The array is "patterned" such that on any one line or column, there is one and only ...
-1
votes
0answers
29 views

limit for a continuous function

If we have a function as follows , then we have to find the value of f(0) so that the function become continuous http://i.stack.imgur.com/oE7TH.jpg For that i tried to find the limit when x ...
0
votes
2answers
25 views

An exhaustion of $C_b(\Omega)$

Consider the space $\Omega=\mathbb{R}^{\mathbb{N}}$ and the space $C_b(\Omega)$ consisting of all bounded continuous functions defined in $\Omega$. Actually we are considering in $C_b(\Omega)$ the ...
0
votes
2answers
46 views

Determining exact value of $\cos (A+B)$ in a specific quadrant

The question reads: Angles $A$ and $B$ are obtuse angles in quadrant 2 (II). If $\csc A = 3$ and $\tan B$ = -1/3, determine the exact value of $\cos (A+B)$. How would I take on this question? ...
1
vote
3answers
65 views

Solving a trig equation that is quadratic?

I have to solve for $x$ given $$\tan^2 x = 2 + \tan x\;\;\;\;\;\;0≤x≤2\pi$$ I brought it all to one side and set it all equal to zero like: $$\tan^2 x - \tan x - 2 = 0$$ What am i supposed to do ...
0
votes
0answers
17 views

Using GPS coordinates in trillateration

for a project we need to find a certain position. The info we have : 3 surrounding positions and the distance between those positions and the point we are looking for. We've got a setup like this ...
0
votes
1answer
18 views

Cardinality of set of functions with coefficients from a set with cardinality omega

Let $A_1$, and $A_2$ be subsets of $\mathbb{R}$ of cardinality $\aleph_0$, $\aleph_1$ respectively. Let $P_1$ be the set of polynomials of the form $a_n(x^n) + a_{n−1}(x^{n−1}) +··· a_1x + a_0$ where $...
0
votes
2answers
46 views

Domain of Integral $\int_{5}^{x} \frac {dt}{(1-t^2)}$

A function reads $$ F(x) = \int_{5}^{x} \frac {dt}{(1-t^2)} $$ Barrons says that the domain of F must be that $x >1$. But why can't $x$ be less than $1$ as well? As long as $x$ does not equal $1$,...
1
vote
3answers
76 views

Evaluating the inverse trigonometric limit $\lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}}$

$$ \lim_{x \to \frac{1}{\sqrt{2}}} \frac{\arccos \left(2x\sqrt{1-x^2} \right)}{x-\frac{1}{\sqrt{2}}} $$ I was doing some questions on limits, I saw one in which there is $\arccos x$. I am stuck ...
-1
votes
1answer
18 views

Limit through a figure

If a circular arc of radius 1 subtends an angle of x radians . The centre of the circle is o and the point c is the intersection of two tangents lines at a and b . Now let T(x) be the area of the ...
0
votes
1answer
20 views

Limit of trig functions

We have to evaluate $$\lim_{x\to 2} \frac{\cos^x a +\sin^x a -1}{x-2}.$$ I am working on it for hours I tried using series , replacing $\cos a$ by $t$ and $\sin a$ by $\sqrt{1-t^2}$ but not got any ...
0
votes
3answers
29 views

Continous function

There are two functions g(x)= ($(2x+1)^{1/2}$-$1$)/x , where x is not equal to zero = 1 , x=0 h (x) = $x^9 - 6x^8 -2x^7 + 12x^6 +x^4 -7x^3 + 6x^2 + x-7$ ...
1
vote
0answers
27 views

Discontinous and differentiable

If we have a two functions $f:[-1/2, 2] \to \mathbb{R}$ $g: [-1/2,2] \to \mathbb{R}$ $f(x) = [x^2-3]$ where $[ \cdot ]$ denotes greatest integer $g(x) = |x| f(x) + |4x-7| f(x)$ Now we have to ...
0
votes
1answer
13 views

Demonstrate that a function is periodic knowing symmetry axis.

Let $$f:\mathbb{R}\rightarrow \mathbb{R}$$ with symmetry axis at $x = {1,2}$. Demonstrate that the function is periodic.
0
votes
2answers
23 views

Number of solutions using graph

We have to find the number of solutions of $e^((-x^(2))/2)$ + $-x^2 =0$ I tried it and got one solutions by drawing graph. Is I have done correct ? My try is on :
-1
votes
1answer
31 views

If the function $f : \mathbb{R} \to A$ given by $f(x) = \frac{e^x - e^{-|x|}}{e^x+e^{|x|}}$ is a surjection, find $A$

If the function $f : \mathbb{R} \to A$ given by $f(x) = \dfrac{e^x - e^{-|x|}}{e^x+e^{|x|}}$ is a surjection, find $A$. I know the fact that "range = co-domain" but was not able to proceed.
-1
votes
1answer
34 views

relation limsup and Continuous function [closed]

Suppose $f$ is continuous and increasing function and $a_n$ is sequence. How can we prove $$f(\text{lim sup}(a_n )) \le \text{lim sup } f(a_n)$$
1
vote
1answer
28 views

What is the slowest growing function that is total but not primitive recursive?

For what I have in mind is the Ackermann-Buck function. If there isn't a slowest growing function do you have examples of other function slower growing than Ackermann-Buck's function?
2
votes
3answers
30 views

Determine the function to be injective when $f: A\to N$, where $A=\{1,4,3\}$ and $f(x)=x^{2}$.

I claimed the function is injective since the elements in domain maps uniquely to elements in the codomain when plug into $f(x)=x^{2}$ . which gives $\{1,16,9\}$ where $\{1,16,9\}$ is found in the ...
2
votes
1answer
20 views

Determine the composition of the functions $f(x)=4x+3$ and $g(x)=-5x^2+1$

Answer: \begin{align*} (f \circ g)(x) & = f(g(x))\\ & = 4(-5x^2+1)+3\\ & = -20x^2+8+3\\ & = -20x^2+11 \end{align*} \begin{align*} ...
0
votes
0answers
23 views

How do I express this function image?

Problem says: if $f(x,y)= (x-y,x,y)$, find $f(\mathbb{R}^2)$. I can see the graphic (thanks to wolfram alpha), but I'm not sure what they are asking me to write with "$f(\mathbb{R}^2)=?$"
4
votes
3answers
141 views

If $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$, then find $f(2)$

Let $f(\frac{x+y}{2})=\frac{f(x)+f(y)}{2}$ and it is given that $f(0)=1$ and $f'(0)=-1$, where $f'$ denotes first derivative. Find the value of $f(2)$ Could someone tell me how to use $f'(0)=-1$ ...
1
vote
1answer
35 views

Distribution for function

I would like a good book to study distribution or generalized functions like the "Basic idea" of that Wiki page. Is there anyone could give me some good book references in this domain? Thanks!
0
votes
1answer
18 views

ordered set notation in functions

Do please forgive me, if this question is a duplicate. How does one correctly notate a function $f$, which takes a ordered subset $S$ from the field $\mathbb{K}$ and returns an other (ordered) subset ...
0
votes
0answers
29 views

Derivative atan2 of a function

I am not able to understand how to solve my doubt. I need to do the : $\frac{\partial}{\partial p} atan2({\cos(\alpha)},{\sin(\alpha)})$ I will compute $\cos(\alpha)$ and $\sin(\alpha)$ as: $\cos(...
0
votes
0answers
36 views

Do we have $f^{-1}(F)=E$ when f is function with domain $E$ and range $F$

Let $f$ be a function with domain $E$ and range $F$ ($f\in F^{E}$ ), Do we have $\mbox{ Inverse image of } F : f^{-1}(F)=E$ Yes, Indeed My Proof 1: we have by definition of function $f\in F^{E}...
0
votes
1answer
42 views

Is this the correct way to translate this phrase into symbols?

The domain of g is the set of all real numbers $x$ such that $x$ is not equal to $-3$. $$g(x)=\{x:\mathbb R|x\ne -3\}$$
7
votes
3answers
104 views

Is really $f(x)=\int g(x) dx$ a function?

I saw many of this kind of questions on some text/question books. Is there any other explanation of this, or is it really wrong as I thought? Here is a question of that kind: If $\displaystyle f(x)=...
0
votes
2answers
22 views

Equal functions with non-equal definitions

Suppose we have two functions $f, g: \mathbb{D} \rightarrow \mathbb{R}$ where $\mathbb{D} \subseteq \mathbb{R}$. Also, $\forall x \in \mathbb{D}: f(x) = g(x)$. My question is: Is it possible for such ...