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

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2
votes
2answers
75 views

Fourier Series of what appears to be a sawtooth series

Find the Fourier series of \begin{equation} f(x)=\begin{cases} x-[x] \quad &\text{if $x$ is not an integer} \\ \frac{1}{2} \quad &\text{if $x$ is an integer} \end{cases} \end{equation} ...
2
votes
1answer
30 views

Two questions about functions

Can someone give me an example of an instance where the pre-image of a function would NOT just be the domain. For instance, for $f(x)=x^2$, the image is all positive reals. The pre-image would ...
0
votes
1answer
68 views

One to one, onto and invertible functions

So I'm trying to see if this makes sense. A function $f:\Bbb R^2 \rightarrow \Bbb R^2$ is said to be invertible if the determinant is different than zero. If it is invertible then it is one-to-one ...
0
votes
2answers
245 views

Proving that hyperbolic sinh is bijective

I have to prove that sinh is bijective. So first i try to profe that it is 1-1: $f(a)= f(b) => a=b$ I will use proof by contradiction: let f(a)= f(b) and $a$ doesn't equal to $b$. After i ...
1
vote
1answer
664 views

Proving a multi variable function is injective

For a function $f : N \times N \rightarrow N$, such as $f(x,y)=2^{x-1}(2y-1)$ how would you go about proving the function is injective? While I understand how to go about proving a function is ...
0
votes
1answer
60 views

Limit of function $f_n(x) = \frac{1}{1+x^n}$ for $x \in [0,1]$

I have this function, and I want to take the limit of it. On the open interval (0,1), the limit is 1, but on the closed interval, when you fix $x$ to be equal to 1, then the limit turns into ...
0
votes
2answers
49 views

Set theory and function composition

$$f = \{ \langle 1,1 \rangle , \langle 2,3 \rangle , \langle 3,2 \rangle \} $$ What does the following composite mean: $$f \circ f$$
-1
votes
2answers
5k views

Writing an equation for a log function given the graph

I have the following graph for a logarithmic function $f$: I don't know any thing about writing an equation for a logarithmic function by knowing it's graph. All what I know is how to draw a graph ...
1
vote
1answer
124 views

Showing Uniform Convergence of $f_n(x) = \frac{nx}{1+nx}$ for $x \geq 0$.

We've been going over Uniform convergence in my class, but I'm very uncertain as to how to effectively show the correct proof. The $\lim_{n \rightarrow \infty}f(x) = 1$, understood using L'Hopital ...
1
vote
2answers
63 views

cardinality of a set of natural lattice points versus natural numbers

Given a set D = $\{(a,b)∣a,b ∈ \mathbb{N}\}$ where L is the set of all points in the first quadrant whose coordinates are natural numbers. Which has more elements, D or $\mathbb{N}$? I know it has ...
1
vote
3answers
54 views

“how to prove this function is one to one ”

Let $f(i,k)=(2^i)(3^k)$ where $i,k$ are natural numbers. Show that $f$ is one to one. I have tried to solve it. Please, can you help me solve it?
1
vote
2answers
134 views

How do you calculate this sum $\sum_{n=1}^\infty nx^n$? [duplicate]

I can not find the function from which I have to start to calculate this power series. $$\sum_{n=1}^\infty nx^n$$ Any tips?. Thanks.
0
votes
1answer
42 views

Problem in understanding solution to a Question?

The solution to one of the question in my test book is as followed $$\tan^{-1}(x^2 - 18x + a) > 0$$ $$x^2 - 18x + a > 0$$ $$(18)^2 - 4a < 0 \implies a> 81$$ problem is that i am ...
0
votes
0answers
40 views

Expressing matrix as its orthogonal

This relates with my question in Proving sum of two matrices to be identity. Given $m<n$ and non singular $n\times n$ symmetric matrix $A$. Suppose that $H$ and $K$ be $m \times n$ and $n\times ...
1
vote
1answer
52 views

Diagnolization of a non-invertible matrix?

Let $A$ be a $n \times n$ matrix. Let $\mathcal{B}$ be a basis for the subspace formed by the columns of $A$. Can there exist a diagonal matrix $C$ such that: $$Cx_{\mathcal{B}} = ...
3
votes
3answers
94 views

Do these matrices exist?

Say you have some non-zero vector in $x$. Can you have two matrices $A$ and $B$ such that: $$Ax = Bx$$ if $A$ and $B$ aren't the identity matrix? This isn't homework, just curiosity.
2
votes
2answers
68 views

Can someone help with this limit? How?

I know what is the solution, but I don't know how to calculate it without l'Hôpital's rule: $$\lim \limits_{x \to 1 }\frac{\sqrt[3]{7+x^3}-\sqrt{3+x^2}}{x-1}$$
1
vote
1answer
41 views

function question

$ f(x) = \begin{cases} -1 && \text{for}-2\le x \le0\\ x-1 && \text{for } 0<x\le2 \end{cases}\\\text{Domain of }f(x) = [-2,2]\\ \text{Let there be be another function g(x),such ...
1
vote
1answer
134 views

Giving injective formulas for a function f: $\mathbb{N} \rightarrow L$ given a set $L = \{(a,b)\mid a,b ∈ \mathbb N\}$

Let $L = \{(a,b)\mid a,b ∈ \mathbb N\}$ $L$ is the set of all lattice points in the first quadrant (all points in first quadrant whose coordinates are natural numbers make $L$) a. Give a formula ...
0
votes
1answer
41 views

Minimum value of $ f(x) = x\log_2x +(1-x)\log_2(1-x) $ [closed]

What is the minimum value of the following function for $ 0<x<1 $ ? Here the base of logarithm is 2 . $ f(x) = x\log_2x +(1-x)\log_2(1-x) $
1
vote
2answers
161 views

“A Function Can't Be Odd&Even” They said, Right? [duplicate]

When I was wondering about if Constant Functions were even or odd, I thought about the function: f(x) = 0 , It's simultaneously odd and even, f(3) = 0 , f(-3) = 0, f(1) = 0 , -f(1) = 0, (It's ...
2
votes
2answers
964 views

Prove $f(x,y) = xy/(x^2 + y^2)$ is continuous everywhere except $(0,0).$

I'd just like to ask you if my proof here is valid. I'll provide you with the method I used and if it seems ok let me know! If not, explanations would be helpful! My main approach to this question ...
2
votes
1answer
38 views

Question about homomorphism of groups

$G,H$ are groups, $\varphi:G\to H$ is homomorphism. How do I prove: $$\ker\varphi=\left\{{e_G}\right\} \Leftrightarrow \varphi\;\text{is injective} $$ I have problem with $\Rightarrow$ direction. ...
1
vote
2answers
43 views

Prove that $ f(A) \subseteq B \implies A \subseteq f^{-1}(B) $

Let $ A,B ⊆ \mathbb R \ $ and $f\colon\mathbb R \to \mathbb R$ be a function. Then $ f(A) \subseteq B \implies A \subseteq f^{-1}(B) $.
1
vote
2answers
101 views

Notation and terminology for functions, interpreting $f(y)$

It seems to me there are two different interpretations of a symbol $f(y)$. I will explain what I mean: Suppose I have a function $f(x) = x$. (I took the identity map to have a simple example). Also ...
5
votes
1answer
217 views

Prove that $f$ is a constant function

Let $f:\mathbb{R}\to\mathbb{R}$ be a function. Suppose: $$\left|\sum_{k=1}^{n}3^k(f(x+ky)-f(x-ky))\right|\leqslant 1\quad\forall n\in\mathbb{N}\quad\forall x,y\in\mathbb{R}$$ Show that $f$ is a ...
2
votes
2answers
64 views

Give the formula for a function with natural domain $(\infty ,0]\cup [10,\infty)$ and with range $(-\infty, -1]$

As the title says, I need to create/generate a formula for a function with natural domain $(-\infty ,0] \cup [10, \infty )$ and with range $(-\infty , -1]$ where $-\infty $ means going off towards ...
0
votes
1answer
65 views

the domain and range of $g(x) = f(6x-4)+8$ when $f(x)$ has a domain: $\{-3,-1,0,2\}$ and range: $\{2,5,7\}$

As the title says, I am trying to find the domain and range of $g(x) = f(6x-4)+8$ when $f(x)$ has a domain: $\{-3,-1,0,2\}$ and range: $\{2,5,7\}$ What would/how do you find the domain of $g(x)$? and ...
0
votes
1answer
73 views

Real Analysis: Convex

$a\geq0$ and $b>0$. Show that $\phi(t) = (a+bt)^p$ is convex on $[0, \infty)$ for $1\leq p< \infty$. Can I just take the derivative and show that is increasing like a calc 1 problem. Or can I ...
1
vote
0answers
47 views

Maximal value of an infinite set. [duplicate]

Consider a continuous function $f$ over an interval $[a,b]$. Let $S$ be the set of all values that $f(x)$ takes over $I$. Intuitively speaking, I believe this set has a maximal and minimal value. Is ...
2
votes
1answer
44 views

Could anyone clarify this definition?

In Michael Spivak's Calculus to explain limits, he uses such an example: We're given a function $f(x) = \left\{0, \quad \quad \quad \quad x \,\,\, \text{irrational} \\ ...
1
vote
2answers
58 views

Range of function

Given the function, $y=f(x)=\frac3{2-x^2}$, find its domain and range. The domain is of course = $R - \{-\sqrt2,\sqrt2\}$. However, the range I got was wrong(rather incomplete). Rewriting the function ...
0
votes
1answer
25 views

Nonincreasing function: the basics

$f(x)$ is a nonincreasing function of $|x|$, does this automatically imply that $f$ is symmetric or even? I haven't seen it written this way. Sorry for asking such basic questions.
1
vote
2answers
53 views

What is the difference between these two functions?

$$r(x) = \langle x, x^2-1 \rangle$$ $$f(x)=x^2-1$$ Their graph is the same, but one is called vector valued function while the other one is a regular one. I think I'll never get to understand this ...
2
votes
3answers
78 views

What is a coordinate shifting?

I need to find the limit: $$\lim_{(x,y,z)\to(1,3-1)}\frac{(x-1)(y-3)+(z+1)^2}{(x-1)^2+2(y-3)^2+3(z+1)^2}.$$ A hint written below says: ...
1
vote
1answer
21 views

Why do they use only the output values to plot the points?

I know that a vector-valued function produces two values from one. Consider f(x)=(2x,5x) Why do they use only the output values to plot the points, whereas at a normal function we use the input as ...
0
votes
1answer
62 views

Solve equation with logarithm

Let $f(x)$ be some distribution function. Let $a\in \mathbb{R}$ and $b>0$. Find $a$ and $b$, such that $$ \ln f(x)=a+bf^{1/2}(x), $$ in addition, it is known that if $f_*(a,b)$ is a solution for ...
1
vote
0answers
24 views

What can I use as the generic term for “a function that is composed with another”?

Suppose I am talking about the composition $g \circ f$ (or more generally $f_n \circ \cdots \circ f_1$). Is there a generic term for the functions $f$ and $g$ (the functions $f_i$)? "Compositand"?
2
votes
3answers
112 views

Example of functions that grow faster than the exponential functions and/or factorial functions?

What is example of functions that grow faster than the exponential functions and/or factorial functions?
3
votes
3answers
157 views

Is this function injective / surjective?

A question regarding set theory. Let $g\colon P(\mathbb R)\to P(\mathbb R)$, $g(X)=(X \cap \mathbb N^c)\cup(\mathbb N \cap X^c)$ that is, the symmetric difference between $X$ and the natural ...
0
votes
1answer
121 views

If two compositions are bijective, then all functions involved are bijective?

Given functions $f:A\to B$, $g:B\to C,$ and $h:C\to D.$ Provided $g\circ f$ and $h\circ g$ are bijective, prove each of the functions $f$, $g$, and $h$ is bijective.
0
votes
2answers
110 views

$n$ dimension integrals involving one dimension Dirac delta functions

I want to calculate an integral $$\iint_D f(x,y)\delta(g(x,y))\,dx\,dy$$ or simply $$\iint_D \delta(g(x,y))\,dx\,dy$$ $\iint_D \,dx\,dy$ is the area of $D$, and$\iint_D \delta(g(x,y))\,dx\,dy$ ...
0
votes
2answers
33 views

for the function $\frac{3x^2+4}{2x-1}$ find the $x$ and $y$ intercepts

With this function I am unsure how to find the $x$ and $y$ intercepts. I am aware that to find the $x$ intercepts I need to put $y = 0$ and vice versa. I have attempted finding the $x$ intercepts ...
2
votes
4answers
687 views

is the empty set a relation?

Is the empty set is a relation? I wonder if the empty set is a relation.in enderton's a relation is a set of ordered pairs. If yes it's a relation why is that?. There is an example in the text for a ...
0
votes
2answers
102 views

Find the domain of the function $f(x)= 3x^2+4/2x-1$ (If they exist)

I'm having trouble understanding how to find the domain of the above function. I've done $2x-1=0$ and got $x=1/2$ but I'm not entirely sure where to take it from there. Is that right? any advice ...
1
vote
1answer
38 views

Monotonicity of a function

Can you guys help me with this question? For which $t \ne 2$ is the function $$f(x) = \frac{(x+t)}{(x-t)}$$ strictly monotonically decreasing in $x_0 = 2$?
0
votes
1answer
40 views

Converting Sum of Trigonometric Functions into Product

I know that $\sin x-\sin y=2\sin(\dfrac{x}{2}-\dfrac{y}{2})\cos(\dfrac{x}{2}+\dfrac{y}{2}).$ I would like to know how to get this formula.
1
vote
2answers
76 views

Prove that the series converges to the integral

Prove: $\int _0^{1}x^{-x}dx$ = $\sum_{n=1}^\infty\frac{1}{n^n} $ I thought of using: $x^{-x}$ = $e^{-x lnx}$ and then using : $e^{-xlnx}$ = $\sum_{n=1}^\infty\frac{(-xlnx)^n}{n!} $ but I'm stuck from ...
2
votes
2answers
59 views

Dual of a sequence

Let $S$ be the set of all sequences $(a_1,a_2,\ldots)$ of non-negative integers such that (i) $a_1 \ge a_2 \ge \ldots;$ and (ii) there exists a positive integer $N$ such that $a_n=0$ for all $n \ge ...
2
votes
1answer
104 views

Why isn't $\log(-1)$ defined?

Why isn't $\log(-1)$ defined. It can be defined as being equal to $i\pi$. Why don't we define the $log$ function over Complex Numbers as well.