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

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6
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2answers
112 views

Intuition/How to determine if onto or 1-1, given composition of g and f is identity. [GChart 3e P239 9.72]

9.72. $A,B$ are nonempty sets. $f: A \rightarrow B$ and $g: B \rightarrow A$ are functions. Suppose $g \circ f = $ the identity function on $A$. (♦) Are the following true or false? $1.$ $f$ ...
0
votes
2answers
56 views

Real Analysis; injective and surjective functions

Let $f$ and $g$ functions from $\mathbb{R}$ to $\mathbb{R}$ given by $$g(x)= x^2-x$$ and $$f(x)= -\sin x$$ i) Is $g$ injective? ii) is $g$ surjective? iii) is $g$ invertible? iv) is $f$ injective? ...
2
votes
1answer
36 views

Suppose a function $f : \mathbb{R}\rightarrow \mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$.

Suppose a function $f : \mathbb{R}\rightarrow\mathbb{R}$ satisfies $f(f(f(x)))=x$ for all $x$ belonging to $\mathbb{R}$. Show that (a) $f$ is one-to-one. (b) $f$ cannot be strictly decreasing, and ...
0
votes
1answer
16 views

Get number of occurences containing a specific number in combinations of N digits?

If I have all the combinations of 3 digits (000 to 999) I want to count how many results contain the digit 4: 456 104 404 ... For 4XX there are 100, for X4X it ...
0
votes
2answers
48 views

Proving that $f$ is a bijection from $N$x$N$ to $N$.

I am having trouble with the following problem: $f: N\times N\rightarrow N$ and $f(i,j)=2^{i-1}(2j-1)$. Prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically equivalent. Work: I ...
1
vote
2answers
31 views

Proving that function with domain (-1,1) is injective.

Function $g\colon (-1,1) \rightarrow \mathbb R$ is defined by $g(x)=\dfrac{x}{1-x^2}$. Prove that $g(x)$ is injective. Work: I shifted the equation so that it ends up like ...
1
vote
2answers
47 views

Proving that a function from $N\times N$ to $N$ is bijective.

I am stuck on this problem: Define $f: N\times N \rightarrow N$ by $f(i,j)=\frac{(i+j-1)(i+j-2)}{2}+i$. How do you prove that $f$ is a bijection thus $N\times N$ and $N$ are numerically ...
0
votes
0answers
37 views

Proving that the set of irrational numbers is uncountable [duplicate]

Work: Assume that the set of irrational numbers is countable. Since $Q$ is infinite, it is therefore denumerable. Therefore, there exists a bijective function $f: N \rightarrow Q$. From here I am ...
0
votes
0answers
12 views

Domain of function of form $f(x)=\frac{g(x)}{k(x)}$

I just want to know did we have a rule to find the domain of function in form of $f(x)=\frac{g(x)}{k(x)}$ .I know $k(x)\ne 0$ . but in general do we have any rule to compute domain of function like ...
1
vote
0answers
21 views

Groups - Compositions

If the f is written to the right of its argument does that mean the composition of $f g$ is actually $g(f(x))$ instead of being $f(g(x))$ which is the notation I'm used to. I ask this because I read ...
0
votes
0answers
45 views

Pointwise convergence and uniform

There is a norm $\Vert .\Vert$ in the space $C([a,b],\mathbb{R})$ such that convergence pointwise implies convergence in norm $\Vert .\Vert$ ? I think not because if there would be the natural ...
0
votes
3answers
40 views

inverse of a function f(x)..change x and y

Find the inverse of the function $f(x)= (2x-1)/(x^2-1).$ we switch the x and y letters and then solve the the equation...but it became kind of complicated while solving
1
vote
1answer
44 views

Proving the chain rule by first principles

I'm currently trying to prove: $(f(g))'(a)=f'(g(a))*g'(a)$ I have been given a proof which manipulates: $f(a+h)=f(a)+f'(a)h+O(h)$ where $O(h)$ is the error function. However, I would like to have a ...
1
vote
1answer
40 views

Working out the median of a beta function

I am trying to work out the median of the beta function of $\mathrm{B}(1/2,1/6)$. I have been told the answer to this is $0.9510$ but i'm unsure to get there? Is there a simple formula in order to get ...
0
votes
1answer
24 views

Figure out simple joint formula for sets of vectors?

I have on the left side four pairs of sample values that result in the respective pair of values on the right side. ...
0
votes
0answers
6 views

Non-monotonic function but Homothetic function

Is it possible for a function to be non-monotonic, but still homothetic? Thank you for your explanations.
1
vote
1answer
71 views

Grade 11 functions [closed]

I need help with three questions for my homework. Any answers would be appreciated. Please try to explain steps or just show them, as I would like to know how to do them. Thanks. Determine if the ...
1
vote
1answer
36 views

Proving that $f: N\times N \rightarrow N$ is surjective [duplicate]

I am having trouble proving that the function $$f: N\times N \rightarrow N, \ \ f(i,j)=2^{i-1}(2j-1)$$ is surjective. Work: I know that using the theorem in which $n$ is the product of prime numbers ...
0
votes
0answers
25 views

Bijective function with different domain and co-domain element count

To be bijective is to be both injective and surjective. Which in other words, have to have a one-on-one match right? Then how am I supposed to come up with a bijective function if the domain has a ...
1
vote
1answer
38 views

Need helping proving that something is differentiable but not continuously differentiable

I need some help please proving that a function is differentiable at $(0,0)$ but not continuously differentiable at $(0,0)$. The function is as follows... (from $\mathbb{R}^2$ to $\mathbb{R}$) ...
1
vote
1answer
26 views

Effect on roots of function on taking the derivative of the function

Suppose there is a function $$f(x)=(x-1)^{15}(x-2)^{20}(x-3)^{25}(x-4)^{30}$$ As we take the derivatives of the function, what will happen to the number of real roots and the number of distinct real ...
1
vote
2answers
29 views

Determining if $Z$ is injective or surjective. Help starting a proof

I have $\mathfrak P(\mathbb{R})$ being the set of all subsets of $\mathbb{R}$, meaning $\mathfrak P(\mathbb{R}) = \{X|X\subseteq \mathbb{R}\}$. I then have $F$ being the set of all functions ...
1
vote
3answers
84 views

What does “$f(x,y)$ is strictly increasing in each argument” imply?

Say we have a function $f(x,y)$. below are what we know about $f(x,y)$ strictly increasing in each argument. $x$ and $y$ are natural numbers only, i.e., $0, 1, 2, ...$ Now we have a fixed number ...
2
votes
5answers
643 views

Explanation of recursive function

Given is a function $f(n)$ with: $f(0) = 0$ $f(1) = 1$ $f(n) = 3f(n-1) + 2f(n-2)$ $\forall n≥2$ I was wondering if there's also a non-recursive way to describe the same function. WolframAlpha tells ...
-1
votes
0answers
34 views

Uncountably infinite: the set of all infinite binary strings [duplicate]

Given that $S=\{0,1\}^{ \mathbb{N} } $ is the set of all infinite binary strings. Is it possible to find a bijective function $f:S\rightarrow \mathcal P(\mathbb N)$? Thank you.
1
vote
1answer
41 views

Prove that $f$ is a bijection

$f : \mathbb{N} \cup \{0{}\} \to \mathbb{Z}$ $f({}n) = \frac{n{}}{2}$ if $n$ is even $f(n) = -\frac{n{}+1}{2}$ if $n{}$ is odd I want to prove that $f$ is a bijection, and find $f^{-1}$. Now I ...
8
votes
4answers
403 views

Is $\tan\theta\cos\theta=\sin\theta$ an identity?

A friend of mine, who is a high school teacher, called me today and asked the question above in the title. In an abstract setting, this boils down to asking whether an expression like "$f=g$" is ...
0
votes
1answer
15 views

Increasing rate of a continuous function

Consider $f: X \rightarrow X$ continuous, with $X \subset \mathbb{R}^n$ compact convex. I am wondering on conditions on $f$ so that there exists $\epsilon > 0$ such that $$ (x-y)^\top \left( f(x) ...
0
votes
1answer
25 views

Property of Projection Operator

Let $p: \mathbb{R}^n \rightarrow X$ be the projection mapping onto $X \subset \mathbb{R}^n$ compact convex. I am wondering if $$ \left( p(x) - p(y) \right)^\top z \leq \left( x -y\right)^\top z $$ ...
0
votes
1answer
65 views

Number of surjective functions from $\{1,2,…,n\}$ to $\{a,b,c\}$

Ok so following questions are given in my text book Let $A = \{1, 2, 3,...., n\}$ and $B =\{a, b, c\}$ then the number of functions form $A$ to $B$ that are onto is. I have no idea how to find ...
0
votes
1answer
52 views

How to solve for $f(2)$ given $3f(x)+f(2-x) = 2x^2$?

I just came across the problem where I'm given that $3f(x)+f(2-x) = 2x^2$ and I need to find $f(2)$. I know it is simple, please help me.
0
votes
1answer
27 views

The difference between semicontinuity and hemicontinuity.

For a point-to-set function F, is "upper hemicontinuous" the same as "upper semicontinuous"? If not, then what's the difference?
0
votes
1answer
29 views

Is there a way to compute if(i < j) k := (a + b)c with polynomials over $\Bbb{Z}_p$?

Let $p$ be a prime and let all variables be in $\Bbb{Z}_p$. Then you can write the result of if(i > 0) k = (a + b)c; (C code) as a polynomial $k := ...
-1
votes
0answers
21 views

Suppose f:A to B. Suppose that C subset A and D subset B. Prove or give a counter example: f (c) subset D iff C subset f^-1(D) [closed]

Suppose f:A to B. Suppose that C subset A and D subset B. Prove or give a counter example: f (c) subset C iff C subset f^-1(D)
1
vote
1answer
35 views

Isomorphism between rings

Let $R$ be the ring of real valued continuous functions defined on the interval $[0, 1]$. Let $I = \left\lbrace f \in \mathbb{R} : f^2(0) + f^2(1) = 0 \right\rbrace$. 1) Prove that $I$ is an ideal. ...
1
vote
0answers
30 views

When does $f_{\omega+1}$ catch up the $G_n$-sequence?

Which is the minimal number k, so that $f_{\omega+1}(n) > G_n$ is true for all $n\ge k$ ? For the definition of $f_{\omega+1}$ look at wikipedia fast growing hierarchy $G_n$ is defined by ...
1
vote
0answers
38 views

How many omegas are there in $\large f_{\epsilon_0}$?

For a description look at fast growing hierarchy at wikipedia. $\large f_{\epsilon_0}$ is not defined any more, it is a power tower of omegas, but how many omegas ? I found a defition $$\large ...
2
votes
1answer
60 views

Prove where $|x|^2(\sin(\pi|x|))^2$ (piecewise) is differentiable in $\mathbb{R}^2$

List all points in $\mathbb{R}^2$ at which $f$ is differentiable as well as ALL points in $\mathbb{R}^2$ where $f$ is not differentiable (implied by the first list) when \begin{equation} f(x) = ...
4
votes
3answers
49 views

Concept of a function and Idea of a formula as a function; History of

Enderton Elements of Set Theory, p. 43 (1977, Academic Press), writes: There was a reluctance to separate the concept of a function itself from the idea of a written formula defining the function. ...
0
votes
1answer
15 views

showing that if a function is a bijection, then there exists a an identity function

Let f:x-y be a bijection, show that foi =iof =f where i is identity function. I know that a bijection is one which is bith noe to one and onto. The problems is that the question is so trivial that I ...
0
votes
2answers
28 views

How do you shift a sigmoidal curve to the right?

How do you shift the function $1$ $/ ( 1 + e ^ {-x} )$ to the right without altering the shape of the curve?
2
votes
0answers
15 views

Quick way to determine the number of horizontal asymptotes

I understand how to calculate horizontal and vertical asymptotes, both by using the trick of comparing the degrees of the numerator/denominator and by using calculus. What I would like to know is ...
0
votes
2answers
24 views

Inverse Function: unique?

Is it true in general that the inverse of a function is unique if it exists? Why is this so? Clearly inverses in groups are unique. However, that seems not directly applicable in this case...
0
votes
2answers
68 views

Solve the following equation for real $y$

Ok, here is another problem I also need help with. Solve the equation for real y: $$2\sqrt[3]{(2y-1)} = y^3 +1$$ This is done by defining $$f(y) = \frac{(y^3+1)}{2}$$ So, the equation becomes ...
4
votes
2answers
297 views

Commutativity of iterated limits

The following is a weird result I've obtained with iterated limits. There must be a flaw somewhere in someone's reasoning but I can't discover what it is. The problem is that, in general, iterated ...
1
vote
1answer
491 views

Hermite identity help

Let $a,b\in\mathbb{Z}$ and $m\in\mathbb{Z}_{>1}$ Evaluate $[\frac {b}{m}] + [\frac {(b+a)}{m}]+ [\frac {(b+2a)}{m}]+ [\frac {(b+3a)}{m}]+ [\frac {(b+4a)}{m}]+ [\frac {(b+5a)}{m}]+.....+ [\frac ...
2
votes
2answers
21 views

Prove that this function is injective

I need to prove that this function is injective: $$f: \mathbb{N} \times \mathbb{N} \to \mathbb{N}$$ $$f: (x, y) \to (2y-1)(2^{x-1})$$ Sadly, I'm stumbling over the algebra. Here is what I have so ...
0
votes
0answers
12 views

what is the use of tilde in specifying domain of functions

what does the tilde "~" in (-3,-2)~{-2.5} mean with respect to domain? when we generally say that the function has its domain from A to B, and excludes C, we can write it as (A,B)-{C}. my ...
0
votes
2answers
21 views

Finding the Asymptote / Root of a reciprocal function

$$y = \frac{3}{8x - 3} $$ The y-intercept is $-1$ and the vertical asymptote is $x = \frac{3}{8}$ but what would be the horizontal asymptote and the x-intercept in this case? I am asking this as the ...
0
votes
2answers
21 views

Rules for combination of odd vs even functional equations

Let $f$ be an even function, and $g$ odd. Let $h$ be some arbitrary function. Is it the case that $f(x) + h(x),\ fh(x),\ hf(x),\text{ and }f(x)h(x)$ are each even or odd according to $h$, and that ...