Tagged Questions

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

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0
votes
1answer
16 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 $f([1,4])$.
3
votes
4answers
52 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
19 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
vote
2answers
30 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?
0
votes
0answers
16 views

A doubt in Kreyszig's Functional analysis

As an application of the Uniform boundedness Theorem, it is proved in Kreyszig's Functional analysis that "There exist real valued continuous functions whose Fourier series diverge at a given point". ...
0
votes
1answer
21 views

Exercises about function composition, bijection and inverses.

So I'm gonna have a test about: Definition of a function, of a surjective and injective function, inverse function, proof that if a function has na inverse, then the functions is bijective, etc The ...
2
votes
1answer
32 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
19 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.
0
votes
0answers
10 views

Holder continuity and gradient

I am trying to prove the implication of differentiability and constancy from Holder continuity. I have: $\frac{\left\lvert f(x)-f(y) \right\rvert}{x-y} \le M|x-y|^{\lambda} \implies \exists g:x ...
1
vote
2answers
39 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
24 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?
1
vote
2answers
47 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
votes
2answers
32 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 ...
0
votes
3answers
50 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
vote
1answer
15 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 ...
2
votes
1answer
55 views

How can I write a function like this? [on hold]

I need to write down and use a function which looks like this. It is some kind of a sinus function. I've no idea how this function looks like, and that is the reason I am looking for your help. Thanks ...
0
votes
2answers
15 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
49 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 ...
1
vote
5answers
46 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
61 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
61 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
17 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
votes
1answer
56 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
votes
1answer
20 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
votes
1answer
40 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
33 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
44 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
votes
2answers
20 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 $a_n$?
0
votes
2answers
32 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 $0$: $$f(x)=\begin{cases} ax+b & x < 0 \\ x−x^2 & x \geq 0 \end{cases}$$ ...
1
vote
2answers
21 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?
0
votes
2answers
31 views

How to find if $k(x)=x^{2}\sin(\pi/2)$, $k(0) = 0$ is differentiable at 0? [on hold]

I need to find whether $$k(x)=\begin{cases} x^2 \sin \frac{\pi}{2} & x \neq 0 \\ 0 & x = 0 \end{cases}$$ is differentiable at $x=0$ or not.
0
votes
0answers
10 views

Angel function and continuity

I have the function $w:\mathbb{R}^2\backslash\{0\}\rightarrow\mathbb{R}$ given by $\cos(w)=\frac{x_1}{||x||_2}\text{ and }\sin(w)=\frac{x_2}{||x||_2}$ after some manipulation I got ...
0
votes
4answers
44 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
votes
1answer
14 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_____ ...
-2
votes
0answers
35 views

Are all single-valued functions bijections? [on hold]

Are all single-valued functions bijections? If not, please explain why.
2
votes
4answers
48 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
78 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
13 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 ...
1
vote
2answers
40 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 ...
1
vote
1answer
20 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 $Ha=Hb$, ...
2
votes
0answers
29 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
29 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
17 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
51 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?
0
votes
0answers
8 views

Prove differentiability, squeeze theorem, using one-sided limits.

This is part of a larger problem, which is f(x) <= g(x) <= h(x), with f(xnull) = h(xnull), f and h are differentiable at xnull, and we eventually show that g is also differentiable at xnull. But ...
-1
votes
2answers
41 views

How to show that $f(x) = x|x|$ is differentiable at 0?

So I've gotten $$f'(x)=\dfrac{2x^2}{|x|}$$ How to show that the following function is differentiable at 0?
0
votes
2answers
15 views

What is the Term for the Center of Mass Equation Structure

What is the term for the generic structure of this form of equation: SUM(Mi * Xi) / SUM (Xi) It is the same as the center of mass calculation.
-2
votes
1answer
23 views

Determine how real parameters a,b,c are ordered [on hold]

We are told that the real-valued function $f(x) = \frac{(x-a)(x-b)}{(x-c)}$, defined except where $x=c$, will assume all real values. Can we say what is the relationship between a, b, c? E.g. is $a ...
0
votes
0answers
20 views

Determine whether a composition of functions is differentiable (prove)

Determine whether a composition of following function is differentiable $$f(x)=\sqrt{x+\sqrt{x+\sqrt{x}}}$$ Just hints, please! Thank you so much!
0
votes
1answer
18 views

An $\Bbb{R}\to\Bbb{R}$ function with two plateaus of different heights and a valley

I am looking for a $\Bbb{R}\to\Bbb{R}$ function $f$ with two plateaus of different heights and a valley. The function has a minimum for $x=a$ and $f(a)=b$. The first (the one for smaller $x$) ...