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

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2answers
102 views

Finding a counterexample to a function proof

This is my proof: If f and g are surjective, then g ◦ f is surjective, with f: A $\to$ B and g: B $\to$ C. I have successfully proved this, but now I have to disprove the converse by finding a ...
1
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1answer
24 views

max and minimum qudratic function problem

A piece of wire $20$ metres long is cut into $2$ pieces and each piece is bent to form a square. Determine the length of the two pieces so that the sum of the areas of the two squares is a minimum. ...
1
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1answer
265 views

Problems Proving Injectivity and Surjectivity

I have these two functions, in which I have to prove or disprove they are injective and surjective: $f:[0,\infty) \to (0,\infty)$ by $f(x) = \frac{1}{x+1}$. $h:\mathrm R \to \mathrm R$ by $h(x,y) = ...
4
votes
1answer
64 views

Why do we have trigonometric functions besides $\sin(x)$?

Probably a terrible question, but I've been curious and can't come up with a reason besides convenience for myself with my limited knowledge. Why do we have $\cos(x)$, $\tan(x)$, etc. when all of ...
4
votes
2answers
68 views

Is $g : \mathbb R →\mathbb R$, $g(x) = |x|$ one-to-one and onto?

So, here is my function, in which I am to prove or disprove both if it is onto and one-to-one: Define $g : \mathbb R →\mathbb R$ by $g(x) = |x|$. For onto, can I say that it is not, because if we ...
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2answers
78 views

Proof of uniform continuity of a rational function

Let $f\colon [0, \infty) \to \mathbb{R}$ be defined by $f(x)= \dfrac{4x}{1+x}$. Show that $f$ is uniformly continuous on $[0, \infty)$.
3
votes
2answers
353 views

Circle to circle homotopic to the constant map?

How to prove that a continuous function, homotopic to the constant map $f:S^1\to S^1$ (a) has a constant point and that (b) $f$ maps $x$ to its antipodal point $-x$?
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2answers
55 views

finding the inverse function of $f(x)=x+\frac{1}{x}$

find the inverse function of $f:\Bbb{R} \to \Bbb{R}$ where $f(x) =x+\frac{1}{x}$. I have tried raising to the power of $2$ but it did not work.
3
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2answers
382 views

Mapping the open ball to itself?

How to prove that there exists a continuous function $f:B^2 \to B^2$ without constant points? Here, $B^2$ is the unit open ball. I guess $f$ can be for example like this $f: re^{iax} \to re^{ibx} $ ...
0
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1answer
46 views

Project a function on a space?

The problem I'm solving is $\begin{cases} & \dot{x}_{1} = -u \\ & \dot{x}_{2} = 4x_{1} \end{cases}$ $x_{1}(0) = x_{2}(0)=0, |u| \leq 3, t \in [0;2], u^{0}(t)\equiv0, J[u] = -2x_{1}(2) + ...
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0answers
39 views

How can a function with asymptotes be defined as a mapping?

A mapping takes each element of a set S and associates it with an element t in some other set T. I believe functions to be mappings. Yet we happily call such as $\frac {x^2}{x+1}$ a function, even ...
5
votes
2answers
314 views

Let : $X \to Y$ be a function. Show that if $f$ is injective then $f(A \cap B) = f(A) \cap f(B)$ for sets $A \subseteq X$ and $B \subseteq X$.

Let : $X \to Y$ be a function. Show that if $f$ is injective then $f(A \cap B) = f(A) \cap f(B)$ for sets $A \subseteq X$ and $B \subseteq X$. My answer : Suppose $f$ is injective and $f(x) \in ...
0
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1answer
33 views

How to find $f^{−1}([9,0])$ and $f([1,4])$ for $f(x)=x-6\sqrt{x}$?

$f$ is a the function defined by $$\eqalign{ f\colon& \Bbb R &\rightarrow \Bbb R_+\\ & x&\mapsto x-6\sqrt{x} }$$ Find $f^{−1}([-9,0])$ and $f([1,4])$.
4
votes
4answers
135 views

Finding inverse of a function $h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$

I have a function: $$h(x) = \frac{1-\sqrt{x}}{1+\sqrt{x}}$$ With just pen and paper, how can I determine if there exists an inverse function? Am I supposed to sketch it on paper to see if it can ...
3
votes
1answer
45 views

Specific piecewise-function SAT2 question

Taken from Barron's SAT Math Level 2 prep book: If f(x) = i, where i is an integer such that i ≤ x < i + 1, the range of f(x) is ...
1
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2answers
34 views

Condition on $ a$ and $b$ so that $f(x)$ has a root?

Let $f(x) = ax(1-bx^{2/3})-1$ where $a$ and $b$ are positive. What is the necessary and sufficient condition on $a$ and $b$ such that $f(x)$ has at least one real root?
2
votes
1answer
49 views

The supermum of E

Let $f\ [0,1]\longrightarrow [0,1]$ be increasing function. let: $$E=\{x\in [0,1] \mid f(x)\geq x \} $$ Show that $E$ has a supermum $b$ and that $f(b)= b$. we have $x\leq 1$ since $f$ is ...
-1
votes
1answer
43 views

Prove $|A| \le|C|$ for injection and surjective functions

$A$, $B$ and $C$ are finite sets with $F: A \to B$ a surjection and $G: B \to C$ an injection. Prove $|A| \le |C|$ I could prove it using examples, but not sure how to generally.
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2answers
80 views

If $f'(x)$ has a limit as $x\to x_0$, then the function $f$ is differentiable at $x_0$

I've got a question about mathematical analysis of one-variable functions. Assume that we have a function defined for $x \neq x_0$ as composition/sum/product of differentiable functions and also ...
0
votes
1answer
42 views

How many function A to B satisfied from f(1)=x

What does it mean to satisfy a function A to B from $f(1)=x$ ? Where $$ A=\{1,2,3,4\}\ \ \text{and}\ \ B=\{x,y,z\}$$ The answer should be $3^3$, but why?
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2answers
69 views

Confusion in notation of functions.

Let us consider the following notations for $x \in X,y\in Y ,z \in Z$. $$F(x,y,z)=x^yy^z$$ $$F_x(y,z)=x^yy^z$$ I am clear with former notation , but I saw latter one too , what's the difference ...
0
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2answers
37 views

Let $A$ and $B$ be countable sets. Is there any function $f$ such that a certain condition holds for an uncountable number of functions $g$?

Let $A$ and $B$ be countable sets. Is there any function $f:A\to B$ such that there exists uncountably many functions $g:B\to A$ such that $g\circ f=\operatorname{id}$ but $f\circ ...
1
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3answers
84 views

Come up with this function

Here's a fun math question: Come up with a function where $$ \begin{align} g''(1) &= 0 \\ g(0) &= 0 \\ g'(0) &= 0 \\ g(1) &= 1 \\ g'(1) &= 1 \end{align} $$ I've tried multiple ...
1
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1answer
34 views

Extension Theorem of twice continously differentiable functions?

Is there a theorem which guarantees me that any function $f$ with bounded first and second order derivatives defined over a compact interval of $\mathbb{R}^2$ can be extended to a twice continously ...
1
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2answers
50 views

Function bijective proving.

Let $\mathbb{C}$ be the set of all complex number. $z\in \mathbb{C}$ Given a function $$ f : \mathbb{C} \to \mathbb{C} $$ $$f(z) = (1+2i)z+5i$$ Prove that it is bijective. First, prove ...
0
votes
2answers
80 views

Closeness of set for not everywhere continuous function

I have a function $w:\mathbb{R}^2\backslash\{0\}\rightarrow\mathbb{R}$ where $w(x)\in[0,2\pi)$. I am also given for free that $w$ is continuous on $\mathbb{R}^2\backslash\{(x_1,0)\mid x_1\ge0\}$. I ...
0
votes
4answers
75 views

Linearity of a function.

I am requested to determine wether these functions are linear or not; to do that, I've to verify both the necessary conditions that are: $f(x+y) = f(x) + f(y)$ $f(\alpha x) = \alpha f(x)$ Now, my ...
4
votes
2answers
90 views

Existence of a differentiable function $f$ such that the set of points at which $|f|$ is differentiable is not dense in $\mathbb R$

Does there exist a differentiable function $f:\mathbb R \to \mathbb R$ such that the set of points at which $|f|$ is differentiable is not dense in $\mathbb R$ ?
4
votes
3answers
89 views

$f:\mathbb R \to \mathbb R$ be twice differentiable , $f(x)+f''(x)=-xg(x)f'(x) , g(x) \ge 0 , \forall x \in \mathbb R$ , then $f$ is bounded?

Let $g:\mathbb R \to [ 0,\infty)$ be a function and $f:\mathbb R \to \mathbb R$ be a twice differentiable function such that $f(x)+f''(x)=-xg(x)f'(x) , \forall x \in \mathbb R$ , then is it true ...
0
votes
1answer
27 views

Prove $|A| \leq |B|$ for $1-1$ function.

Prove $|A|\leq |B|$ if function $F:A\rightarrow B$ is a $1-1$ function. I wanted to know how to prove this out of curiosity. The help is appreciated.
0
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1answer
64 views

Prove that $f(x) \in A$ if and only if $x \in f ^{−1} (A)$.

Is there even a proof for this or is this just by definition : $f(x) \in A$ if and only if $x \in f^{−1}(A)$.
0
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1answer
37 views

Proving: If a function is bounded, then the fuction's limit is bounded.

The question I have to answer is the following: Let I be an open interval that contains the point c and suppose that f is a function that is defined on I except possibly at the point c. If $m \le ...
0
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1answer
53 views

A question on logic and some functional inequalities

Suppose that I have a (generic) function $g$ and arguments $a, b \in \mathbb{N}$. I know that $g$ satisfies the inequalities $$1 < \frac{g(b)}{b} < \frac{g(a)}{a} < 2.$$ I also know that ...
0
votes
2answers
54 views

Is the bijectivity of a function equivalent to monotony and continuity?

My high-school math professor told us that in order for a function $ f $ to have a reverse it must be monotonic and continuous, but I always thought that necessary and sufficient condition for a ...
0
votes
1answer
46 views

What's the name of this function?

Does the function $f(x)=\log(-\log(x))$, $x\in(0,1)$ has a name? Equivalently, the function $g(y)=f^{-1}(y)=\exp(-\exp(y))$, $y\in{\mathbb R}$. The only thing I want to know if whether this function ...
0
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2answers
34 views

Increasing function non-continuous on points of sequence - construction

How to construct strictly increasing function $f$, non-continuous on points of countable sequence of numbers $a_n$?
1
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5answers
142 views

How to find for which real numbers $a$ and $b$, the following functions are differentiable at $0$?

I need to find for which real numbers $a$ and $b$, the following functions are differentiable at $0$: $$f(x)=\begin{cases} ax+b & x < 0 \\ x−x^2 & x \geq 0 \end{cases}$$ ...
0
votes
2answers
34 views

Intersection of trig function

There are two trig function graphs on the same set of axis. $f(x)=\sin(2x)$ and $g(x)=\cos(3x)$. How do I go about finding the points of intersection of the two graphs?
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3answers
78 views

Number of One to One Functions [duplicate]

Suppose a set A has n number of elements and a set B has m number of elements. Then why the number of one to one functions is n!? And also, how many functions in total are possible? Are they n*m? I ...
0
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1answer
23 views

Finding the y-vertex of a function and X2.

I am trying to solve the following exercise: The graph of the fuction $y=-2x^2+bx+c$ passes through the point (1,0) and has as its vertex the point (3,S). What is the value of s? Options: A -5_____ ...
1
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4answers
59 views

What function produces {0, -8, 8, -16, 16, … }?

I'm trying to figure out a function that produces the set of numbers {0, -8, 8, -16, 16, ... } when given the set of positive integers. I'm having a hard time understanding what makes some results ...
3
votes
0answers
86 views

Functional inequalities involving cubing and incrementing

Consider the set $S$ of positive increasing invertible functions $f$ satisfying: $$f((x+1)^3-1)≤(f(x)+1)^3-1$$ $$f(x^3)≥(f(x))^³$$ $$f(x)+1≤f(x+1)$$ for all positive real $x$. Clearly the identity ...
0
votes
0answers
24 views

Generic way to find codomain of a function

Is there a generic way (an algorithm maybe?) to find a codomain of a function, if the domain of all constituents is known. I.e., I have an editor where users can write simple expressions (by using ...
3
votes
4answers
525 views

Can a continuous real function take each value exactly 3 times? [duplicate]

Let $f: \mathbb{R} \to D \subseteq \mathbb{R}$ be a continuous function. Is there a function $f$ that satisfies the following property? $\forall y \in D$, there are exactly 3 $ x_1,x_2,x_3 \in ...
1
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2answers
49 views

How do I convert this parametric expression to an implicit one

I have: $$x=5+8 \cos \theta$$ $$y=4+8 \sin \theta$$ With $ -\frac {3\pi}4 \le \theta \le 0$ If I wanted to write that implicitly, how would I do it? I get that it's a circle, and I can easily write ...
2
votes
1answer
82 views

Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function.

Let $H$ and $K$ be normal subgroups of a group $G$, with $H \subseteq K$. Define $\phi:G/H \rightarrow G/K$ by $\phi(Ha)=Ka$. Prove: $\phi$ is a well defined function. [That is, if $Ha=Hb$, ...
1
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0answers
44 views

Function with $f(x)f(y)=f(xy)$ satisfying intermediate value property

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(xy)=f(x)f(y)$ for all $x,y\in\mathbb{R}$, and $f$ satisfies the intermediate value property. Taking $x=0$, we have $f(0)=f(x)f(0)$. ...
1
vote
1answer
37 views

Continuity of a function at $0$

A similar has been asked before, but it was confusing. Please help me with it. I need a general method of dealing with such problems I need to show that the following function is continuous at $0$. ...
0
votes
1answer
137 views

Find the range of a complicated function

I need to find the range of the following function : $$f(x,y) = \sqrt[4]{\frac{4x - 3y + 5}{3y-4x + 13}}$$ So my thoughts about it are first the bottom part $( 3y - 4x + 13 )$ must be greater than ...
2
votes
1answer
63 views

Weird function or not

Is $f\colon\emptyset \to\mathbb{R}$ with $f(x) = (-1)^{\frac{1}{2}}$ a function where $\emptyset$ is the empty set and $\mathbb{R}$ is the set of real numbers?