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

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0
votes
1answer
67 views

General form of iterates $f^{n}(x)$ for $f(x) = \frac{x}{1-x}$

Let $f: \mathbb R\to\mathbb R$, $f(x) = \frac{x}{1-x}$. Define $f^{2}(x) = f(f(x))$, $f^{3}(x) = f(f(f(x)))$, ... Guess the form for $f^{n}(x)$ and prove your answer is correct using ...
2
votes
0answers
27 views

Finding the image of $f$ given by $f(x)=\frac {2x^2-x}{3x^2+x+1}$ for $\forall x\in [0,\infty)$

I found the image of $$f(x)=\dfrac {2x^2-x}{3x^2+x+1}$$ for all $x\in \mathbb R$ using the method of $f(x)=b$ then checking when the discriminant is positive I got that it's positive between $-1/11\le ...
0
votes
1answer
27 views

Getting rid of a fractional power over f(x)

I start with the following relation. $\frac{dy}{\sqrt{y}} = -h dt$ I then integrate it and get this function. $y^{\frac{3}{2}} = -\frac{3}{2}ht + C$ My algebra is rusty, so I'm stuck at this ...
0
votes
1answer
72 views

Is it true if $g\circ f$ is injective then $g$ is injective?

I need to know if this statement is true or false: If $g\circ f$ is injective then $g$ is injective. I couldn't try to prove this statement. I was thinking to show a counter-example: ...
-2
votes
1answer
127 views

No injection between the power set of a set A and the set A. [duplicate]

For finite sets it's easy to prove it because the cardinal of the power set it's bigger than that of the set so there won't be enough elements in the codomain for the function to be injective. What is ...
2
votes
4answers
200 views

Determine the existence of a function

Does there exist a function $f:\mathbb{R}^n \to \mathbb{R}$ such that for all $x = (x_1,...,x_n) \in \mathbb{R}^n$: $f(A) = f(x)$ for every permutation A of $\{x_1,...,x_n\}$. $f(x + (a,...,a)) = ...
2
votes
1answer
36 views

How to find the range of the function $\frac{e^x log_{e} x 5^{x^2+2} (x^2-7x+10)}{2x^2-11x+12}$

How to find the range of the function $$\frac{e^x log_{e} x 5^{x^2+2} (x^2-7x+10)}{2x^2-11x+12}$$ We can see the domain of the function is $(\frac{3}{2}, 4) \cup (4, \infty)$ as the denominator is ...
2
votes
1answer
145 views

Is every injection between sets of the same cardinality also a bijection?

I have to decide if the following statement is true or not: For all sets $A$ and $B$ of the same cardinality, if a function $f: A \to B$ is injective, then it is also bijective. I would say that ...
1
vote
1answer
51 views

Determine whether each function is one-to-one, onto, or both

$g:\mathbb Z \times \mathbb Z$ where $g$ is defined by $g(x)=x-1$ My guess is that this is onto and one-to-one. But is the correct interpretation of this problem that $g$ is a function of an ...
0
votes
2answers
29 views

Finding a domain of a function

What would be the best approach finding this function's domain? $$f (x) = \sqrt{\cfrac{x-2}{x+2}} + \sqrt{\frac{1-x}{\sqrt{1+x}}}$$ Can I just calculate the domain of each expression seperately? $$ ...
3
votes
1answer
53 views

Ways of defining a recursive function that counts right-parenthesis in a string

I'm trying to find a more elegant way of defining a recursive function on $\{(,)\}$ that counts right-parenthesis in a string. Let r be a function on $\{(,)\}$ defined recursively so that: ...
0
votes
2answers
38 views

A problem in real valued function on compact set. [duplicate]

If $f$ be a real valued continuous function defined on $[0,2]$ such that $f(0)=f(2),$ then prove that there exist a $ x \in [0,1]$ such that $f(x)=f(x+1).$ I tried in the following way, Since $f$ ...
3
votes
2answers
175 views

Is the greatest common divisor injective? Is it bijective?

In an examination paper, there were the following questions: Is gcd an injective function? Is gcd a bijective function? I found these questions odd because I thought that we need to ...
0
votes
2answers
31 views

Function + Differentiation

Given the function $f(x) = ax^3+bx^2+cx+d$. Determine the value of $a$, $b$, $c$, and $d$ knowing that the curves passes through points (-1,2), (2,3) and that the tangents at the points on the curve ...
0
votes
1answer
37 views

Pre-image under a invertible function same as image of the inverse function?

Say $f:X\to X$ is continuous and bijective. Then is the pre-image of a set $I\subset X$ under the function $f$ the same as the image of $I$ under the inverse function $f^{-1}$?
1
vote
1answer
128 views

Inverse of $f(x)=\dfrac{x}{4-x^2}$?

I tried writing $x=\dfrac{f^{-1}(x)}{4-[f^{-1}(x)]^2}$ but can't make $f^{-1}(x)$ subject of the formula.
0
votes
1answer
20 views

semi continious functions characterizations

Does anyone knows how to prove this: Let $f: (X, d) \rightarrow \mathbb{R}$ be an upper semi-continious function. Prove that $f$ is u.s.c. if and only if $ \{ x \ \ |\ \ f(x) \geq z \} $ is closed ...
2
votes
1answer
350 views

Simple formula for a sieries like 1, 2, 5, 10, 20, 50, 100, …

I'm looking for a simple formula that will give a series that looks like this: $1; 2; 5; 10; 20; 50; 100; ...$ That means a function that will give this output: $f(1) = 1$ $f(2) = 2$ $f(3) = 5$ $f(4) ...
0
votes
2answers
46 views

Do there exist non-constant function f, g for which the “naive quotient rule” holds?

A common mistake students make is applying a naive form of the quotient rule to functions of the form f/g, mistakenly applying the product rule and arriving at f'g+g'f for the derivative. What I'm ...
1
vote
1answer
73 views

How do I find the horizontal asymptote of $f(x)=\frac{\sin (x) }{x}$?

I can instantly see that there will be a vertical asymptote at $x=0$, however I am finding it quite a challenge to find a horizontal asymptote. I've drawn the graph and it seems as if the amplitude of ...
0
votes
2answers
28 views

A problem in vector valued function

Let $f:R^2 \to R^2$ be defined by $f(x,y)=(x+y,xy).$ I intend to show that inverse image of each element in $R^2$ under f has at most two elements. that is the possibilities for the number of ...
0
votes
1answer
206 views

Proving a bijection with absolute value

let $g(x) = \frac{x}{1-|x|}$ for all $x \in (-1,1)$ a) show that $g$ is a bijection from $(-1,1)$ to $\mathbb{R}$ Here is what I have done. Say $g(x)=y$ so $$y = \frac{x}{1-|x|}$$ iff ...
3
votes
1answer
55 views

Finding where $|x^2-1|$ is increasing and where not.

Consider the function $f: \Bbb R \to \Bbb R$, given by $f(x) = |x^2-1|$. Then, the exercise asks to find $X,Y \subset \Bbb R$ such that $X \cup Y = \Bbb R$, $X \cap Y = \varnothing,$ $f_{\large| X}$ ...
3
votes
2answers
86 views

Is there a continuous bijection from $\mathbb{Q}$ to $\mathbb{Q^*}$

I am trying to find an explicit function but I don't know if that is even possible. Thanks
0
votes
2answers
40 views

Inverses / Bijections

Let $f:A\to B$, and $g:B\to A$ such that $$ g(f(a))=a \ ,\ \forall\ a \in A, $$ and $$ f(g(b))=b\ ,\forall\ b \in B. $$ Does this mean that $f,g$ are inverses and bijections? Bests
0
votes
2answers
153 views

how to prove that which constants a,b,c,and d it is true that f o g = g o f

im working on a functions unit and Im stuck on this problem: Let $f(x) = ax + b$ and $g(x) = cx^2 + dx$ (a,b,c,d are constant). Compute f o g and g o f. And determine for which constants a, b, c and ...
0
votes
2answers
98 views

Determine for which constants $a$, $b$, $c$ and $d$ it is true that $f \circ g = g \circ f$.

Let $f(x) = ax + b$ and $g(x) = cx^2 + dx$ where $a,b,c,d$ are constants. Determine for which constants $a$, $b$, $c$, and $d$ it is true that $f \circ g = g \circ f$. I have the equations for $f ...
0
votes
1answer
89 views

Focus of a parabola, without derivatives

I have a seemingly easy question, but I have no clue how to find out its answer. I have the function $$f(x)=\tfrac{1}{8} x^2$$ This function is for (a parabolic cross-section through) a paraboloid ...
1
vote
1answer
43 views

Is the function one-to-one?

Is the function $f: \mathbb R+ \to\mathbb R \\$ defined as $f(x) = \sqrt{x} + x + 2$ one-to-one? I'm pretty sure the function is one to one but when I try to solve $f(x) = f(y)$ to $x = y$ I get ...
2
votes
2answers
39 views

Proving that $f$ is a bijection.

Here is the question: Suppose $f:X\longrightarrow Y$ and $g:Y\longrightarrow X$ satisfy $$\forall x\in X.(g\circ f)(x)=x,\,\forall y\in Y.(f\circ g)(y)=y$$ Prove that $f$ is a bijection, with ...
2
votes
3answers
54 views

Function theory and set theory with cartesian products

Let $S$ and $T$ be sets and define the function $$f:\mathcal P(S) \times \mathcal P (T)\to \mathcal P(S \cup T)$$ by $f(A,B) = A \cup B$ for all $A \subseteq S$ and all $B \subseteq T$. Prove that ...
3
votes
3answers
131 views

Bijection, and finding the inverse function

I am new to discrete mathematics, and this was one of the question that the prof gave out. I am bit lost in this, since I never encountered discrete mathematics before. What do I need to do to prove ...
1
vote
1answer
19 views

Is it correct to say that since the derivative of a function is zero at a certain point imply that this point is an inflection point?

I would like to know if it is correct to say that if a polynomial function P has its derivative that is equal to zero at one point then: There is only one inflection point thus the function has at ...
1
vote
2answers
52 views

if $f(g(x))$ is 1 to 1, then is g 1 to 1?

Im working on practice problems that the instructor gave us yesterday and Im stuck with this question.. the question is: if $f\circ g $ is one to one, then is $g$ one to one? Im not sure how to ...
-1
votes
1answer
32 views

Find $K$ such that $|(x, y)| > K$ implies $(x - 1)^2 + (y + 2)^2 > C+ 4$.

For any š¶ āˆˆ ā„, find š¾ such that |(š‘„, š‘¦)| > š¾ ā‡’ š‘„2 + š‘¦2 - 2š‘„ + 4š‘¦ + 1 > š¶ i.e. (š‘„ - 1)Ā² + (š‘¦ + 2)Ā² > š¶ + 4 whenever |(š‘„, š‘¦)| > š¾ NOTE: š¾ is a function of š¶ only, and does NOT depend ...
4
votes
0answers
37 views

How to read an expression that is ambiguous?

(1) How should I parenthesize $\log n \log \log n$? Also: (2) What general rule/rationale is used to do this parenthesization? To elaborate; I see why $\log\log n$ is unambiguous, but $\log n \log ...
1
vote
1answer
62 views

Does any nice shape have a function describing it?

Consider the standard $xy$-plane. If I draw a curve in the plane that passes the vertical line test, can you find a function $f:\mathbb{R}\to\mathbb{R}$ whose graph is this shape? How about in ...
0
votes
1answer
30 views

What is a common name for the resulting function?

Consider the following population regression model: $$y_{i} = \beta _{1} + \beta_{2}x_{i} + \epsilon _{i},$$ where $i=1,...,n$. Assume $\epsilon \sim iid$, with the pdf in equation: $f(\epsilon ) = ...
0
votes
1answer
53 views

Finding inverse of $g(x) = \dfrac{3x + 1}{2x + g(x)}$

Find $g^{-1}(3)$ given $g(x) = \dfrac{3x + 1}{2x + g(x)}$ My Approach: \begin{align*} y & = \frac{3x + 1}{2x + y} && \text{(does $g(x)$ become $y$ also?)}\\ x & = \frac{3y + 1}{2y ...
1
vote
2answers
387 views

Prove composition of bijections is bijection

Let f : A ā†’ B and g : B ā†’ C be bijections. Prove that gā—¦f : A ā†’ C is a bijection Can someone show me the steps I should take to solve this problem?
0
votes
1answer
153 views

Random distribution: Parabola distribution (non-linear distribution)

Thanks to a very helpful answer, I recently successfully implemented a linear probably distribution (for my open source constraint solver) to select an element out of ...
1
vote
1answer
54 views

What are the possible values of $x$?

For what values of $x$ does this equation holds? $$2\arctan(x)=\arctan\left(\frac{2x}{1-x^2}\right)$$ The answer is $-1<x<1$ Why? How can we say this?
3
votes
1answer
66 views

Injection from $\mathbb{R}^n$ to $\mathbb{R}$

I'm doing some set theory problems. They ask me to find the cardinality of some specific set and it seems that it would often be useful to have a function that mapped any tuple of real numbers to a ...
1
vote
2answers
36 views

Find an open set $B$ such that $g^{-1}(B)$ is not open

I cannot understand part ii) in this solution. I cannot see the significance of arbitrarily close to 0 points for which $|sin(\frac{1}{x_n})|=1$
1
vote
1answer
32 views

Smoothness of a particular function in two variables

I cannot understand why this doesnt work for $v \neq 0$ and $u=0$? I think it may be to do with my lack of understanding of why $v=0$ and $u \neq 0$ works which I believe has come from the fact that ...
0
votes
2answers
18 views

Determine the asymptotic behavior of $f(n)$ in relation to $g(n)$

$f(n)=n^\sqrt{n}, g(n)=2^n$ $f(n)=10^{\log\log n}, g(n)=\log n$ Note: $\log$ is in base 2. For section #1, I tried to evaluate the limit $\lim_{n\to\infty} \frac{2^n}{n^\sqrt{n}}$ but got stuck ...
0
votes
3answers
52 views

How to determine if this function is one-to-one

I have a question about how to determine if the function is one to one The question to the problem is, Is $f:\mathbb{R}^+ \rightarrow \mathbb{R}$ defined as $f(x)= \sqrt{x}+x+2$ one to one? I know ...
0
votes
1answer
59 views

Create an equation for a description of a rational function

A graph has a y-intercept at -5, no x-intercepts, and discontinuous points at (-1,-5) and (3, -5). I want to form an equation for this graph, but I don't know how the y-intercept relates to the graph ...
0
votes
1answer
59 views

Was my inference regarding $u(z)=log(z)$ correct?

I have already solved the problem but would appreciate a clarification in part (b). A has initial wealth $w$ and faces a loss $l$ with known probability $\pi$. Insurance available at unit price ...
1
vote
2answers
61 views

Prove that g(y)>0 for all y in the real numbers

Let g:$\mathbb{R}\to \mathbb{R}$ such that ($i$) for all $y_{1},y_{2} \in \mathbb{R}$, $g(y_{1}+y_{2}$)=$g(y_{1})g(y_{2})$ suppose in addition that ($ii$) there exists $y \in \mathbb{R}$ such ...