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

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0
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1answer
16 views

show a cartesian product in function is injective or surjective?

I had previously figured out injectivity/surjectivity on basic functions but I am stumpted when it comes to showing functions which are cartesian products are injective/surjective. The first one: ...
0
votes
1answer
19 views

Quadratic Functions: Determine the value of b

I'm having trouble with this question and I'm not sure what to do. Would appreciate any one who helps me out. Question: The point $(-2,1)$ is on the graph of the quadratic function: $f(x) = -x^2 + bx ...
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0answers
32 views

A function of class $C^2$

Problem: given a function $F:\mathbb{R}^2\mapsto\mathbb{R}$ of class $C^2$, with $F(0,0)=0,\nabla F=(2,3)$, shown that a surface $F(x+2y+3z-1,x^3+y^2-z^2)=0$ can be given locally at $(-2,3,-1)$ as ...
1
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0answers
32 views

Question concerning defining a particular class of functions

I have a multiset of real numbers $X \subseteq \mathbb{R} $ and I want to create a class of injective function to map the elements of $X$ to the unit interval(so basically a normalization). However ...
-3
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1answer
69 views

If fg is surjective, then g is surjective.

Either prove or give a counterexample to the "converses" of exercise 2 on page 17 If fg is injective, f is injective. If fg is injective, g is injective. If fg is surjective, f is surjective. If fg ...
1
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2answers
43 views

Partial derivative in two dimensions

I am struggling with section 3.3 of the following thesis https://smartech.gatech.edu/xmlui/bitstream/handle/1853/29610/grigo_alexander_200908_phd.pdf. Page 21 is fine, then the problems occur in ...
2
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1answer
34 views

How many of these are surjective?

Let $A=\{a,b,c,d\}$ and $B=\{e,f,g\}$. How many maps are there from A to B? How many of these maps are surjective? $\textbf{Part 1:}$ There are 4 elements in A and 3 elements in B. Thus there are ...
1
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0answers
17 views

If $y=f(x)$ is a linear function satisfying the relation $f(xy)=f(x)f(y)$, then the curve $P(x,y)=\alpha$ cuts $y=f^{-1}x$ at?

If $y=f(x)$ is a linear function satisfying the relation $f(xy)=f(x)f(y)\forall x,y\in\mathbb R$, then the curve $$y^2+\int_0^x(\sin t+a^2t^3+bt)dt=\alpha,\alpha\in\mathbb R^+$$ cuts $y=f^{-1}x$ ...
-2
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0answers
28 views

For a boat to float in a tidal wave the water must be 2.5 meters deep… (trig questions)

$$y=5+4.6\sin\left(\frac{t}{2}\right)$$ What is the period in hours? Simply $p=2\frac{\pi}{n}=4\pi$ which is $\pi$ per hour If the boat leaves the bay at midday what is the latest time it can return ...
2
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1answer
61 views

If $f \circ g$ is surjective, $g$ is surjective

If $f \circ g$ is surjective, $f$ is surjective. If $f \circ g$ is surjective, $g$ is surjective. $\textbf{Part 1:}$ Let $f:B \to A$ and $g:C \to B$. Assume $f \circ g$ is surjective. Since ...
2
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5answers
226 views

If Bob and Alice never met in class, at least one of them missed at least half of the classes

Alice opened her grade report and exclaimed, "I can't believe Professor Jones flunked me in Probability." "You were in that course?" said Bob. "That's funny, i was in it too, and i don't ...
1
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1answer
35 views

How many points to span a goniometric wave and how to construct the goniometric function

I have two questions concerning the spanning of a simple trigonometric function: What is the minimum number of points to define/span a "simple" trigonometric wave in two dimensions? Is it possible ...
0
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1answer
25 views

Determining if a function decreases exponentially

Define a function: $f(x) = \sqrt{\frac{e^{-kx}}{1-e^{-kx}}}$ where $k > 0$. Does this function decrease exponentially? EDIT: Sorry, I meant to ask just if it decreases exponentially.
-3
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1answer
28 views

Calculus net signed area [closed]

so this homework problem I have asks me to find the net signed area. So, what I did was 1+2+3+4+5+6-5-4-3-2-1 = 6, but that is wrong... Why is it wrong? Thanks!
0
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1answer
53 views

To find the inverse of an implicit function

I have a function $t(f)$ here: $t(f) = T(sin(2\pi f/B)/2\pi + f/B) $ for $[-B/2 \le f \le B/2]$. $B$ and $T$ are constants. How to find the inverse of this function that is $f(t)$ using numerical ...
1
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1answer
48 views

Function - Main Features?

I understand how to draw this function, but what does it mean by main features? any examples for the question below? Consider the function $f : \mathbb{R} \rightarrow \mathbb{R}$ given by $f(x) = ...
3
votes
2answers
70 views

Find the generating function of the sequence $a_n = \sum\limits_{k=0}^n k(k-1)$

Find the generating function of the sequence $ a_n =\sum\limits_{k=0}^n k(k-1)$ My try: Let's assume $k(k-1)$ is genereated by $F(x)$ then $a_n$ is generated by $\frac{F(x)}{1-x}$ (that's a ...
1
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2answers
32 views

Form of Function of Two Variables

Let $V(Q,T)$ be a function of two variables. The exact functional dependence is not known, but it is known that: $$V(Q,T)=f(Q)T,$$ and $$V(Q,T)=g(T)Q.$$ How do I prove rigorously that $$V(Q,T)=cTQ,$$ ...
1
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3answers
33 views

Simple inverse function of $\frac{1-2x}{1+x}$

Just started learning about inverse functions, and got stuck on this one: $$f(x) = \frac{1-2x}{1+x}$$ So I tried multiplying by $(1+x)$ on both sides and got $y+yx = 1-2x$ but that doesn't seem to ...
0
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5answers
67 views

Showing that $f(x) = \ln x - e^x$ has no real roots

Show that $f(x) = \ln x - e^x$ has no real roots Since $\displaystyle\lim_{x \to 0^+} f(x) = -\infty$ and $\displaystyle\lim_{x \to \infty} f(x) = \lim_{x\to \infty} e^x \left ( \frac{\ln x}{e^x} ...
0
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1answer
23 views

Help explain the set being constructed in this Cantor-Schroder-Berstein proof

The Cantor-Schroder-Bernstein theorem states that: Suppose $A$ and $B$ are sets. If $|A|\le |B|$ and $|B|\le |A|$, then $|A|=|B|$ Proof: So, $|A|\le|B|$ implies we can choose an injection ...
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2answers
56 views

Not understanding the answer to the inverse of a function

$$f(x)\quad =1+\sqrt { 1+x } $$ $$y\quad =1+\sqrt { 1+x } $$ $$y^{ 2 }\quad =1+1+x$$ $$y^{ 2 }-2\quad =x$$ How is it $x=y^{ 2 }-2y$ ?
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1answer
46 views

How to prove: if $x = y$, then $f(x) = f(y)$ (for a function $f$)

The intuition of this for me is that if two elements in the domain are equal, then their images are equal. But I don't know if this is actually true or not, so I want to know if this is for sure. ...
2
votes
2answers
72 views

Why can't equations with unknown inside and outside of a function be solved in a standard way?

For example the equation $$ n2^n = 8 $$ Is true for $n=2$ which can be guessed, but more complicated examples would require some sort of approach. Also with trigonometric functions, $$ x\sin(x) + ...
7
votes
1answer
305 views

Can functions have multiple inputs?

Now bear with me here, I'm not the best at math. I'm just trying to find out something that I never really learned. I was wondering, can a function have multiple inputs such as this one below? ...
0
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3answers
34 views

Powerset bijection problem

Please do not provide a full answer for this. Let $2^{S} = \{f : S \rightarrow \{0, 1\}\}$. For $A \subseteq S$, define $\chi_{A}\in2^{S}$ by $$\chi_{A}(s) = \begin{cases} 0 & \text{if } s ...
1
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1answer
21 views

How can I construct a specific sigmoid function?

The simple sigmoid function $$f(x)=1/(1+e^{−x})$$ approaches zero as x tends to negative infinity, and approaches $1$ as x tends to positive infinity. But I want to set $1$ and $20$ instead of $0$ and ...
0
votes
2answers
51 views

Terminology for $1/(e^x+1)$?

$ \frac{1}{e^x+1} $ and $ \frac{e^x}{e^x+1} $ Just wonder if either of the above function has a term/name associated with it? Or they are just functions that look beautiful without names? Maybe they ...
0
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1answer
16 views

function undefined at odd inputs

I am a high-school student in pre-calculus. My teacher told me today that it is impossible to define a function using only multiplication, division, exponents, addition, subtraction such that it ...
0
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3answers
36 views

Making a Piecewise Function a Single Function

Is there a way to turn a piecewise function into one function. For example: $$\ f(x)=\begin{cases} g(x) & \text{if $a≤x<b $} \\ h(x) & \text{if $b≤x≤d$} \end{cases}$$ (Can you use the ...
1
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2answers
49 views

Proving that the given function $F:\mathbb N\times \mathbb N\to\mathbb N$ is bijective

Consider the function $F:\mathbb N\times\mathbb N\to \mathbb N$ defined by $$F(a,b)=\frac{(a+b-2)(a+b-1)}{2} +a$$ How can I prove that it is a bijective function? I proved it using Partial ...
2
votes
1answer
36 views

How to prove this version of the Cantor-Schroder-Bernstein theorem?

My text states the Cantor-Schroder-Bernstein theorem as follows: Suppose that $X$ and $Y$ are non-empty sets such that $|X|>|Y|$. Then, any function $f:X\rightarrow Y$ is not an injection, i.e. ...
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0answers
17 views

continuity of derivative of a differentiable function

Give an example of a function which is differentiable on [a,b] but it's derivative is not continuous on that interval. I already know one: F(x)= {x2.sin(1÷x),x is not equal to zero & x when x=0} ...
2
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4answers
6k views

How to calculate Maximum or Minimum of two numbers without using if?

How to to calculate the maximim or minimum of two numbers without using "if" ( or something equivalant to that manner)? The above question is often asked in introductory computer science courses and ...
0
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1answer
35 views

Finding the slope at two points.

I have been sitting at this for 2 days and I'm not getting anywhere, admittedly I might be just very dumb when it comes to mathematics, and as such I would really appreciate some help with this. I ...
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votes
3answers
47 views

Is $\lfloor -x \rfloor$ the same as $-(1+\lfloor x \rfloor)$?

Provided $x$ is a positive real, does $\lfloor -x \rfloor = -\left(1+\lfloor x \rfloor\right)$ ? If not, is there a relation between relation between $\lfloor x\rfloor$ and $\lfloor-x\rfloor$?
0
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2answers
34 views

Finding asymptotes for $f(x)=\frac{x^2+3x-10}{3x^2+13x-10}$

$$f(x)=\frac{x^2+3x-10}{3x^2+13x-10}$$ I know that the horizontal asymptote is $1/3$. To find the vertical asymptotes, I set the denominator equal to zero and used the quadratic formula, and I got ...
1
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0answers
18 views

Axis of a Paraboloid?

The elliptic paraboloid $$x = y^2 /3 + z^2 /7$$ is a bowl-shaped surface. Along which axis does the bowl open? I don't even know to get started on this question. Thank you.
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0answers
10 views

Give the transformations of the following functions.

Give the transformations of the following 3 functions. Can you please give me at least 3 points to plot for each function(keeping the domain restriction in mind)? Also for rational function. Also ...
7
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1answer
173 views

Determine an explicit expression for $f$.

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ a continuous function, bounded such that the space $\mathrm{lin}\{f_k(x)=f(x+k)∣k ∈\mathbb{N}\}$ is finite-dimensional. Determine an explicit ...
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2answers
45 views

Cardinality of all injective functions from $\mathbb{N}$ to $\mathbb{R}$.

What is the cardianlity of: $$ A = \left\{ f:\mathbb{N}\to\mathbb{R} : \text{f is injective} \right\} $$ Trying to prove it using Cantor–Bernstein–Schroeder theorem, I have the obvious side: $$A ...
2
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2answers
27 views

Proving $\mathbb Z[i]$ is euclidean domain .

From the definition of euclidean domain , one has to select euclidean function . Let $\mathbb Z[i]=\{a+bi | a,b\in \mathbb Z,i=\sqrt{-1}\}$ We have to select an euclidean function $f$ , such that ...
1
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1answer
54 views

Why is the set of functions from $\mathbb N$ to $\mathbb Q$ not equal to the power set of $\mathbb N \times \mathbb Q$?

I'm trying to show that $^{\mathbb{N}}\mathbb{Q}$ is not equal to $\mathcal{P}(\Bbb {N} \times \Bbb {Q})$. I think that I would have to show that either $^{\mathbb{N}}\mathbb{Q} ...
2
votes
1answer
45 views

Which is the property of the functions that correspond to this definition/examples?

I'm looking for a definition for a particular function(-input) property. Considering a function $f$ that takes as input a list of elements and produces in output just one element, which is the ...
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2answers
40 views

Is the relation a function?

Is the relation on $\Bbb {R}$ a function from $\Bbb {R}$ to $\Bbb {R}$? $$\{(a^2,a)\mid a \in \Bbb {R}\}$$ How do I determine whether or not the relation is a funtion? Would I treat $(a^2,a)$ as ...
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0answers
52 views

How to prove the uniqueness of a specific root?

Let us define: $$F(x):=\int_0^Tf(t)\cos(x\,t)dt-\frac{\sin(T_0\,x)}{T_0\,x}$$ where: 1). $0<T_0<T \in\mathbb{R}^+$ are both positive real constants, and 2). $0\leqslant f(x)\in C^{\infty}$ ...
1
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1answer
28 views

Why is this relation a function?

I need to determine whether or not the relation $\{ (a^2,a) | a \in \Bbb {R}, a \geq 0\}$ is a function from $\Bbb {R}$ to $\Bbb {R}$. I think that it is a function. But I don't know how to ...
2
votes
2answers
64 views

Is the relation a function

I'm trying to determine if the relation $\{(\frac{a}{b}, a-b) | a,b \in \Bbb {Z}, b \neq 0\}$ is a function from $\Bbb {Q}$ to $\Bbb {Z}$. I know that a relation is a function from A to B if dom(f)=A ...
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votes
1answer
42 views

Why is the set of functions from the naturals to the rationals a subset of $\mathcal P(\mathbb N \times \mathbb Q)$?

Explain why the set of functions from the naturals to the rationals is a subset of $\mathcal{P}(\mathbb{N} \times \mathbb{Q})$? Give an example to show that the set of functions from the naturals to ...
1
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2answers
513 views

Trying to figure out a formula with given input and outputs.

I'm playing this video game where people can get kills, deaths, and assists , and all this is recorded on a stats website. The stats website gives you a rating by directly manipulating these numbers. ...