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

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0
votes
2answers
31 views

Determine a and b so that function is continious

$$ g(t)= \begin{cases} 2t^2 ;& t<-1 \\ at ;&-1<t<1 \\ bt-\frac 12 ;&t>1 \end{cases} $$ How can I determine $a$ and $b$ so this function $g$ is continuous at whole $\mathbb R$. ...
1
vote
0answers
44 views

Sigmoid function with fixed bounds and variable steepness [partially solved]

(see edits below with attempts made in the meanwhile after posting the question) Problem I need to modify a sigmoid function for an AI application, but cannot figure out the correct math. Given a ...
10
votes
3answers
1k views

A binary operation, closed over the reals, that is associative, but not commutative

I am aware that matrix multiplication as well as function composition is associative, but not commutative, but are there any other binary operations, specifically that are closed over the reals, that ...
-1
votes
3answers
74 views

Domain of the function $\frac{1}{\sqrt {x^{12}-x^9+x^4-x+1}}$

What is the domain of $$\frac{1}{\sqrt {x^{12}-x^9+x^4-x+1}}$$ the answer is $(-\infty,\infty)$. Now the polynomial has degree $12$. Also it's continuously increasing from $1$. So I thought there ...
0
votes
1answer
29 views

Determining if a function is onto

If our range such as in the question below is all the real numbers excluding $0$, to determine if a function is onto we must ask if all real numbers excluding $0$ can be mapped to at least one value ...
-1
votes
0answers
13 views

Resolution function explicity [on hold]

Examine where the equation $f(x,y)=0$ locally by $y=h(x)$ can be resolved. Calculate in all these places $h'(x)$ by implicit differentiation. Enter the resolution function(s) $h(x)$ explicitly if this ...
0
votes
6answers
89 views

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$?

How many solutions exist for the equation $2\sin(x)+\cos(x)=\sqrt{3}$ in $[0,2\pi]$ ? All I could till now : LHS =$2\sin{x}+\cos{x}$ Since, $−\sqrt{5} \leq 2\sin{x}+\cos{x} \leq \sqrt{5}$ So a ...
2
votes
2answers
37 views

A question on mapping inside unit disc

For an analytic function $f:\bar D→\bar D$ where $\bar D$ is the closed unit disc centered at origin.Suppose $\bar D=D\cup$$\delta D$, where $\delta D$ is the boundary of open disc $D$ and $f$ is onto,...
2
votes
1answer
57 views

What are all the different classes of functions upon real numbers and what do they mean, exactly? [closed]

I have been hearing terms like "piecewise C1", "continuous", "linear", "piecewise constant", "trigonometric", "logarithmic", "exponential", "elementary", etc. functions for many years. I know what ...
4
votes
0answers
59 views

Prove that $f$ is invertible

Did I show enough to prove $f$ is invertible? Alternatively is there a more efficient way to do so? Thanks in advance for any help. Let $f : X \rightarrow Y $a nd $g : Y \rightarrow X$ be ...
2
votes
2answers
56 views

If $C\cap D=\emptyset$ Prove that $f^c(C)\cap f^c(D)=\emptyset$

Is this the proper way to go about proving this? By showing $C\cap D$=$f^c(C)\cap f^c(D)=\emptyset$? Any feedback would be greatly appreciated. I don't have any other way of getting feedback for my ...
3
votes
2answers
158 views

Is it ok to use Kronecker delta function to find if one of its variables belongs to a half open interval?

Kronecker "delta" function is generally defined as $\delta(i,j)=1$ if $i$ is equal to $ j$, otherwise $0$. How about if $j$ is not an integer? I mean let $j$ is a half open interval defined as $j=(0,...
-1
votes
2answers
61 views

How to show is function surjective? [duplicate]

$f\left(x,y\right)=\left(\frac{x}{x^{2}+y^{2} } , \frac{y}{x^{2}+y^{2} } \right) $ $R^{2}-(0,0)$ Can someone help me with shwoing that this function is surjective?
0
votes
4answers
73 views

How to find inverse of function?

$f\left(x,y\right)=\left(\frac{x}{x^{2}+y^{2} } , \frac{y}{x^{2}+y^{2} } \right) $ How do I find inverse function? I have trouble here, because there are 2 variables and I am not sure how to do it?...
0
votes
1answer
32 views

Is the function $f: n \in N^{*} \longrightarrow |D(n)| \in N^{*}$ injective/surjective?

Is the following function injective/surjective? $$f: n \in N^{*} \longrightarrow |D(n)| \in N^*$$ (where $D(n)$ is the set of all the divisors of $n$). My attempt: It is NOT injective because $f(2) ...
-2
votes
3answers
48 views

Help with Trigonometric Functions

so while playing around with circles and triangles I found 2-3 limits to calculate the value of $ \pi $ using the sin, cos and tan functions, I am not posting the formula for obvious reasons. My ...
0
votes
1answer
16 views

Limit of succession of function (2 variables)

i want to find limit of $f_n(x,y)=\frac{e^{-ny^2}}{1+x^2+y^2/n}$, for $n\to+\infty$. I think $f_n\to f=\frac{1}{1+x^2}$ because I plotted it via MatLab. But my first idea was $f_n\to0$ because of $e^...
1
vote
0answers
39 views

Integrals of the form $\int x^m(a+bx^n)^Pdx$

I was reading a book on Integral Calculus, and in one chapter, the author dealt with methods of solving Integrals of the form $$\int x^m(a+bx^n)^Pdx$$ The author broke it down into 4 cases:$$$$ $...
1
vote
1answer
68 views

Is the definite integral $\int\limits_a^b {f(x)dx}$ a function of $f(x)$?

I was wondering if definite integrals are functions of their integrands. For example, if $y=\int\limits_a^b {f(x)dx}$, can we call $y$ a function of $f(x)$ ?
0
votes
2answers
44 views

How to find minimum and maximum of function?

Given: the function $$f\left(x,y\right)=4x^{2} +3y^{2} -5x$$ Find the $x$ values of the minimum and of the maximum on the set: $$ \left\{ \left(x,y\right)\in \mathbb R ^{2}: x^{2}+y^{2}=9 \right\}...
1
vote
1answer
30 views

Cardinality of Surjective only & Injective only functions

I'm a college student just beginning to study the very basic of set theory. In studying about Surjective & Injective functions & how they map their domain to their codomain, it came to my mind ...
1
vote
2answers
69 views

Determine $f(x)$ which satisfies given condition

Suppose $f(x)$ is real valued function of degree $6$ satisfying the following conditions: $1.$ $f(x)$ has minimum at $x=0$ and $x=2$ $2.$ $f(x)$ has maximum at x=1 $3.$ $lim(x \to 0)$ $\frac{ln(\...
-3
votes
1answer
46 views

Determine function is onto or not? [closed]

A function is defined as $$f(x) = \frac{e^{x^2}-e^{-x^2}}{e^{x^2}+e^{-x^2}}$$ f is from $\mathbb R\to \mathbb R$. check if function is surjective or not, injective nature of function can be proved ...
0
votes
2answers
50 views

Determining if a power set is one to one or onto.

Let $P$ be the power set of $\{a,b,c\}$. A function $f: P \to \mathbb{Z}$; the set of integers, follows: For $A$ in $P$, $f(A)=$the number of elements in $A$. Is $f$ one-to-one? Explain. Is $f$ onto?...
0
votes
1answer
30 views

Prove for some $n$, $f^{n+1}=f^n$ and that Y is bijective.

Are these sufficient to show what is being asked? If you could confirm or provide a more efficient way to do so I would greatly appreciate it. Let $X$ be a finite set and $f:X\rightarrow X$ be a ...
-1
votes
4answers
69 views

Disprove the statement: If $g\circ f=I_X$then $f\circ g=I_Y$. [closed]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: Let $f : X \rightarrow Y$ and $g : Y \rightarrow X$ be functions. If $g\circ f=I_X$...
0
votes
1answer
23 views

How to prove that a function only defined for integers (or primes, or multiples of 3, etc.) have a certain derivative?

I would like to know how a function not continuously defined (for a lack of better words) could be proven to have a certain derivative (or, if the word "derivative" isn't appropriate, rate of growth). ...
0
votes
0answers
14 views

How to substitute demand parameters from one function into another

Here are two inverse demand functions: $$ p_1= α_1-β_1q_1= 8−2q_1 $$ and $$p_2 = α_2-β_2q_2 = 4−\frac{1}{2q_2}$$ How can you get this:$$ gcs_1(q) = q(8−q)$$ and $$gcs_2(q) = \frac{q(16−q)}{4}$$ from ...
0
votes
2answers
32 views

Name or closed form for given description of a function

I am looking for the name or nice explicit formula for the following function: I give you a positive integer N, and the function I want from you, say, f(X), subtracts X from N until the result is ...
4
votes
4answers
212 views

Is any real-valued function in physics somehow continuous?

Consider the following well-known function: $$ \operatorname{sinc}(x) = \begin{cases} \sin(x)/x & \text{for } x \ne 0 \\ 1 & \text{for } x =0 \end{cases} $$ In physics, the sinc function has ...
1
vote
3answers
82 views

Is function invertible?

Reflection on the unit circle: Let $E=\mathbb R ^{2} - \left\{0,0\right\} $ be perforated plane and $f: E \mapsto E$ defined by $f\left(x,y\right)=\left(\frac{x}{x^{2}+y^{2} } , \frac{y}{x^{2}+y^{2}...
0
votes
0answers
77 views

Normal family theorem regarding meromorphic functions (Schiff, Joel)

I have a question regarding Theorem 3.3.1 from pages 76-77 in Joel Shiff's book Normal Families. The theorem is stated as such: $\textit{Let} \, \, \mathcal{F}$ $\textit{be a family of meromorphic ...
1
vote
1answer
23 views

Production functions total cost

Production function is: $f(L,M)=L^{1/2}M^{1/2}$. L is the number of units of labour, M of machines used. Cost of labour is 9 per unit, whereas the cost of machine is 81 per unit. Total cost of ...
0
votes
1answer
36 views

How to prove the following function is convex?

I was working on a problem and it reduced to show that $$f(a)=log\Big(\sum_{i=1}^{r}a^ix_i\Big)~~a>1, x_i>0$$ is convex. I have $$f^{\prime \prime}(a)=\frac{\partial^2f(a)}{\partial a^2}=\frac{[\...
0
votes
1answer
14 views

How to determine if a function is quasiconcave or quasiconvex using calculus

I would like to know if there is a theorem which links the quasi concavity of a function to the sign of its second order derivative. For eg. we know a function is Concave in a given interval if it's ...
0
votes
1answer
22 views

Problem in equality of two functions.

First of all, my book states that two functions $f$ and $g$ are equal iff.: $Dom(f) = Dom(g)$ $Codom(f) = Codom(g)$ For each $x\in Dom(f)$ , $f(x) = g(x)$ and vice-versa. which is fairly ...
-1
votes
0answers
28 views

Kuratowski's representation for an ordered pair [closed]

Could anyone explain Kuratowski's representation for an ordered pair for me? I wish to know what 'adjoined with' means and how nested sets are made.
2
votes
2answers
38 views

Finding the function of a sine graph that has both translation and transformation

I can't quite find a problem similar enough to this yet, and I need some serious help. Here is a photo of the graph of the function I am trying to find out: Sorry, but I don't have enough ...
2
votes
1answer
38 views

Adding functions when one has undefined point [closed]

If two functions are defined as set of points and there's a point that is defined only in one function but not in the other, e.g. $A = \{(0,1)\}, B = \{(1,2)\}$. In function arithmetic, what would be $...
-5
votes
0answers
27 views

Complete the square [closed]

Research results for a clothing retailer show that 600 people will buy a sweater for 130\$. For every 1\$ increase in price, 25 fewer people will buy the sweater. The function that models the revenue ...
1
vote
0answers
29 views

Motivation behind substitutions in an integral 1

I was reading a textbook on Integration where I came across suggested substitutions for certain types of Integrals. These were as follows: $$$$ Integrals of the form $$\int\dfrac{dx}{(ax+b)\sqrt{...
0
votes
0answers
33 views

How is the Möbius function in boolean sets?

Although the text is a little long, the text is very simple so that're familiar with the matter. I did not understand two passages in the text. Could you help me show that: $I'= I \cup M$ and only ...
2
votes
0answers
26 views

2D Cauchy Distribution Peak [closed]

Is the general form of a 2D Cauchy Peak, if A is the amplitude: $$\frac{A}{1+\frac{(x-x_0)^2}{\gamma_x^2}+\frac{(y-y_0)^2}{\gamma_y^2}}$$ $?$
1
vote
2answers
41 views

Composition of a piecewise function and another function

I have this two functions. $f(x)=\arcsin \left(\dfrac{3-x}{3x-1} \right)$ and $g(x)=\begin{cases} 0 ;& |x| <\pi \\ \sin(2x);& |x| \ge \pi \end{cases}.$ I have to find $f \circ g$. I ...
0
votes
1answer
19 views

Exception to definition of a function.

My book gives me this definition of a function: A function $f$ is a special kind of relation,i.e $f\subset A\times B$,such that the following hold: for each $a\in A$ there exist $b \in B$ ...
1
vote
1answer
53 views

Functional inequality $f(x_1+x_2)\ge f(x_1)+f(x_2)$

Given a function $f$ on the interval $0\le x \le 1$. We know that this function is non-negative and $f(1)=1$. Moreover, for any two numbers $x_1$ and $x_2$ such that $x_1\ge 0, x_2 \ge 0$ and $x_1+x_2\...
0
votes
0answers
21 views

Iterating a relation to find a function

I was playing around with a graphing calculator, trying to find approximations for inverses of $f(x)=x^5+x+1$. This cannot be expressed with radicals or the like, but I wanted to see how close I could ...
3
votes
4answers
111 views

Disprove the statement $f(A \cap B) = f(A) \cap f(B)$ [duplicate]

If someone could walk me through this I would greatly appreciate it. Disprove the following statement: If $f : X \rightarrow Y$ is a function and $A$, $B$ are subsets of $X$ then $f(A \cap B) = f(A)...
1
vote
2answers
49 views

Recursively counting divisors of a number

I want to make a recursive function f that counts all (not only prime) different divisors of a given natural number: $f(n): = |{a ∈ ℕ | ∃ b ∈ ℕ : a . b = n }| $ ; with $ f(0)=0 $ for example $ f(3) ...
-3
votes
0answers
24 views

Prove a primitive recursive function

I'm supposed to prove that the division of integers (whole numbers) is primitive recursive. I know that the add, subs, mult are primitive recursive but I don't know if that helps, I'm not asking for ...