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

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2
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1answer
62 views

Which functions are 1-1 and which are onto?

Let $N$ denote the naturals, $Z$ the integers, $R$ the reals. Which of the following is 1-1? Onto? $(a) f(x) = x^2$ on $N, Z, R$ Suppose $f(x_1), f(x_2) \in N$. Since $x_1, x_2 \in N : x_1 \neq ...
0
votes
2answers
59 views

Why does this limit work?

Let $h(x)= (1+1/x)^x$ and $g(x)$ be another function. Now suppose $\lim\limits_{x \to \infty} g(x)= \infty$. Then $\lim\limits_{x \to \infty} h(g(x))$ =$\lim\limits_{x \to \infty} h(x)=e$. I would ...
2
votes
3answers
288 views

How to show $(gf)^{-1} = f^{-1}g^{-1}$?

Suppose that $f:A\rightarrow B$ and $g:B \rightarrow C$ are both one-to-one and onto. Prove that $gf$ is one-to-one and onto. Prove further that $(gf)^{-1} = f^{-1}g^{-1}$. I have already proven ...
0
votes
2answers
78 views

Prove $f(\cap \scr{C}) \subset \cap f(\scr{C})$. Confused on why it's not a symmetric relation?

If there are any minor mistakes in my proof, it would be great if they were pointed out - but let it not be the central discussion. I'm rather concerned why the answer is $\subset$ instead of $=$ ...
1
vote
1answer
48 views

Prove this statement…

I need to prove that the inverse function of a linear function is also linear? I suppose that a linear function is defined like this: $$f(x)=ax+b$$ but how can I prove that its inverse is linear?
0
votes
1answer
2k views

Proof of an odd function plus an even function

I was looking at the wiki page http://en.wikipedia.org/wiki/Even_and_odd_functions#The_sum_of_even_and_odd_functions and it says that to prove an even function plus an odd function, we first have to ...
0
votes
1answer
27 views

Minimum of function

Let $f:\mathbb{R}\to\mathbb{R}$ be a function with $f(x)=\dfrac{(x^4-2ax^3+3a^2x^2-2a^3x+a^4+9)}{(x^2-ax+a^2)}.$ Determine the minimum of the function, if we know, that $-2\leq a\leq2$, $a\neq0$. I ...
-1
votes
1answer
440 views

Find the sum-of-products expansions

Find the sum-of-products expansions of the Boolean function $F(x, y, z)$ that equals $1$ if and only if a) $x = 0.$ b) $xy = 0.$ c) $x + y = 0.$ d) $xyz = 0$.
0
votes
1answer
41 views

A more elegant version of this function?

I challenged myself. The goal was to find a function $f$ with two variables $x$ and $y$ real, which results $1$ if $x=y$ and results $0$ if $x ≠ y$. But, the fonction can only use additions, ...
1
vote
2answers
43 views

Is there an elementary continuous function which is positive only if all arguments are?

I am looking for a continuous function $f:\mathbb R^n \to \mathbb R$ such that $f(x_1, x_2, \ldots , x_n) > 0$ if and only if $x_i > 0 \ \forall i = 1,2, \ldots , n$. Can anyone suggest a good ...
1
vote
0answers
163 views

Matlab (machine learning) - How to calculate binary scatter plot with straight line distinguishing these values?

Maybe title sounds a bit confusing, so I'll try to explain it. Let say we have a some matrix 4x4 = (0 0 0 1 0 0 0 1 1 0 0 0 0 0 1 0) So some function should ...
1
vote
1answer
340 views

Questions about coercive functions and its implications

Given this definition: A function $f:\mathbb{R}^n\rightarrow \mathbb{R}$ is $coercive$ if $$\lim_{||x||\rightarrow\infty}f(x) = \infty.$$ Explicitly, this means that for any $M>0$ there is an ...
6
votes
2answers
216 views

Does inverse of a nontrivial holomorphic function always have a branch point?

Any nontrivial (i.e. which is not a first order polynomial) entire in $\mathbb{C}$ function I have thought of has a multifunction as its inverse and has a branch point. For example, ...
4
votes
2answers
456 views

How to find the range of $\sqrt {x^2-5x+4}$ where x is real.

How to find the range of $$\sqrt {x^2-5x+4}$$ where $x$ is real. What I've tried: Let $\sqrt {x^2-5x+4}=y$, solving for real $x$, as $x$ is real discriminant must be $D\geq0$. Solving I get ...
0
votes
2answers
65 views

Proof that a function is surjective to R

I'm having difficulties proving that the function $$\frac{\sin(\frac1x)}{x^2}$$ is surjective to $\mathbb R$. on the interval $(0,10]$. I tried to use the intermediate theorem, but that of ...
3
votes
1answer
71 views

Prove/disprove statements regarding continuous functions.

I found some old math tests from my school years and thought it might be fun to see what I still remember. The answer is simply, not as much as I hoped for. I'm having trouble proving/disproving these ...
2
votes
2answers
134 views

Determine the set of points where $f$ is continuous.

Define $f:[0,1]\rightarrow\mathbb{R}$ by $$ f(x) = \begin{cases} x & \text{if $x$ is irrational} \\ p\sin(\frac{1}{q}) & \text{if x=$\frac{p}{q}$, where $p,q$ are relatively prime integers.} ...
3
votes
1answer
214 views

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and ...
0
votes
2answers
88 views

Proof that $f: \mathrm{P}(X) \rightarrow \mathbb{N}$ is injective

Situation: Take $n \in \mathbb{N}$, let $X = \{0, 1,..., n-1\}$ and define map $f: \mathrm{P}(X) \rightarrow \mathbb{N}$ through $f(A) = \sum_{i \in A} 2^i$ Question 1: Proof that $f$ is injective. ...
5
votes
1answer
614 views

Find lower bound of function

Can someone help me finding a lower bound to the function $$f(x)=\frac{x-1}{e^{-1}-xe^{-x^2}},$$ where $x\in[1,+\infty[$? Taking the derivative and then solve $f'(x)=0$ isn't analytically possible. ...
2
votes
1answer
42 views

Making logarithmic function go higher

I am looking at logarithmic functions, and, lets say, log2 (x+3) is having a bit of a growth rate between 0-10 values of ...
2
votes
1answer
94 views

Online logarithm drawing

I am looking for a site that will give me the output of my logarithms. What I want to do, is I want to input, in example log(2), and I want it to draw an output ...
2
votes
1answer
363 views

Help with writing MATLAB code - Consider the function $f(x) = x \cos(-x^2)$

Would anyone be able to help me with this question? Consider the function: $f(x) = x \cos(-x^2)$ Write MATLAB functions f.m and fp.m for the function $f$ and its derivative $f '$, respectively.
0
votes
1answer
58 views

How exactly do I graph piecewise functions?

I am an Algebra 2 student and I am studying piecewise functions. How exactly do I graph some piecewise functions like: $$f(x) = \begin{cases} -x-5 &: x \le -3\\ 4 &: -3 < x < 2\\ ...
3
votes
2answers
55 views

Find the equation with roots, $A$, $B$, $C$ is $ABC=6$, $A+B+C=5$ and $A^2 +B^2+C^2=21$

Find the equation with roots, $A$, $B$, $C$ is $ABC=6$, $A+B+C=5$ and $A^2 +B^2+C^2=21$ Can someone please hint me, or show me what do i do with this question please. Im quite clueless and need to be ...
3
votes
2answers
107 views

A function continuous in both arguments

Is there a two-arguments function which is not continuous but continuous in each argument? It seems I have studied something like this, but don't remember.
6
votes
1answer
83 views

Wrong use of function notation $f(n)$

I've recently read in a book about computational complexity theory: $$ O(f(n)) = \{g:\mathbb N \to\mathbb R \cup \{0\} : \exists \xi > 0,n_0\in \mathbb N\;\: g(n) \leq \xi \cdot f(n) \;\: \forall n ...
3
votes
1answer
44 views

The set of all polynomial functions from $\mathbb{Z}^3 \rightarrow \mathbb{Z}/(2)$

Let $f:\mathbb{Z}^3 \rightarrow \mathbb{Z}_2$ be a polynomial function in $\mathbb{Z}[x_1, x_2, x_3]$. Then $f$ has the form $f(x_1, x_2, x_3) = c_1 x_1 + c_2 x_2 + c_3 x_3 + c_4 x_1 x_2 + c_5 x_1 ...
2
votes
1answer
62 views

Why does the domain of a function such as sqrt(x-5) /sqrt(x+2) change when rationalizing the denominator?

I was tutoring a student the other day and the above function in the title came up. I initially showed her how to get rid of the radical, and then we proceeded to find the domain of the rationalized ...
2
votes
1answer
97 views

Finding the Domain and Range of a function composition

I'm having trouble finding the domain and range of a function composition. $f(x) = x^2 - 3x$ $g(x) = \sqrt{x}$ $(g \circ f)(x) = g(f(x)) = \sqrt{(x^2 - 3x)}$ How do I find the domain and range of ...
-1
votes
1answer
114 views

I'm taking an advanced math paper and I have no idea how to start this question!

How would I go about working this out? I honestly don't know where to start! Any help is appreciated.
0
votes
1answer
27 views

Finding a formula using two other functions.

I am trying to find a formula for $p(x)$, but I am not seeing where to start. I have two functions $m(x)=9*27^x$ and $n(x)=9^x$ I need to find the formula for function $p(x)$ if $m(x) = p(n(x))$ ...
2
votes
1answer
69 views

If $f(f(x,a), b) = f(x, p)\; \forall x, a, b$, is it also true that $f(f^{-1}(f(x,a), b), c) = f(x, q) \;\forall x, a, b, c$?

Let $f:\mathcal{F}\times\mathcal{G} \to \mathcal{F}$ be a function of two arguments, defined over finite sets $\mathcal{F}$ and $\mathcal{G}$, and bijective with respect to the first argument. Let $g ...
0
votes
2answers
387 views

Find the Inverse function of f. $f(x)=1+\sqrt{1+x}$

I found the Inverse of the function, $f^{-1}(x)= x^2-2x$. The back of my pre-cal book gives me the inverse of the function and the domain. What I don't understand is, how the domain comes to be $x ...
6
votes
4answers
126 views

The limit of $f$ or the limit of $f(x)$?

I have read before that $f$ denotes the function $f$ whilst $f(x)$ denotes the value of the function $f$ at $x$. What is right? To say that the limit of $f$ as $x$ tends to $a$ is $L$ or to say that ...
2
votes
1answer
28 views

Question of how to write certain response

What is the domain of the function $$f(x,y)=\frac{1}{x^2+y^2-1}$$ The answer is clear, right?! $$x^2+y^2-1\neq 0\Longrightarrow x^2+y^2\neq1$$What is the point contained in the circle of ...
0
votes
1answer
59 views

Number of symmetric functions in a binary vector space of length n

A function $$f : \{0,1\}^n \to \{0,1\}$$ is called symmetric if for every $x_1,x_2,\ldots,x_n \in \{0,1\}$ and every permutation $\sigma$ of $\{1,2,\ldots,n\}$, we have $$f(x_1,x_2,\ldots,x_n) = ...
0
votes
2answers
951 views

cdf/pmf/pdf validity question

Studying for a statistics exam. I have come across this problem: and it presents to me some important and extremely basic questions (I have a LONG way to go before I'm prepared for this exam). ...
0
votes
1answer
198 views

Showing that $g$ is 1-1 given $f$ onto and $gf$ 1-1?

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be functions. If $f$ is onto and $gf$ is one-to-one, prove $g$ is one-to-one. I want to know if the reasons why I stated my statements are ...
4
votes
1answer
40 views

Given an inductive function, how to calculate?

Currently having slight difficulty figuring out how to solve this. Given is; $$\begin{align}f(0) &= -3\\ f(1)&= 2 \\ f(n) &= f( n - 2 ) + 2 f( n - 1)\end{align}$$ Now, I need to ...
0
votes
0answers
45 views

Find the characteristic equation of a recursive function

I want to determine whether the following recursive function is unstable; $$ x(t+1) = \left( wx+sx(t)^b \over w+x(t)^bs + (1-x(t))^b(d-s) \right) $$ Wikipedia is telling me that I want to have the ...
4
votes
0answers
48 views

Decomposability in the tensor product sense of functions of two variables

Let $S$ and $T$ be "nice" metric spaces, e.g. complete normed fields like $\Bbb R$, $\Bbb C$ or $\Bbb Q_p$. Let $F$ be a function $$ F:S\times T\longrightarrow K $$ where $K$ is a topological field ...
2
votes
1answer
26 views

Is it permissible to locate the abscissa of extreme points of $y=f(x)$ by powering the function first for the sake of simplicity?

Let's take a simple example as follows, $$R=\sqrt{A^2+B^2 +2AB\cos \theta}$$ It represents the magnitude relation of vectors $\vec A$, $\vec B$, and $\vec R$ which is $\vec A +\vec B$. And we have ...
0
votes
3answers
78 views

Why does the function f(|x|) look like this?

If the image is f(x), why does f(|x|) look like two triangles above the x axis (basically the right side duplicated on the left)?
0
votes
2answers
122 views

For what values of $z$ is $f(z) = e^z$ real? Imaginary?

I feel like I might understand this already, but I just wanted to make sure. I said that $f(z)$ is real is $z$=any real number and $f(z)$ is imaginary if $z=rj$, where $r$ is any real number. Thanks.
1
vote
3answers
65 views

How to show if $gf$ is 1-1, $f$ is 1-1?

Let $f: A \rightarrow B$ and $g: B \rightarrow C$ be functions. If $gf$ is 1-1, prove $f$ is 1-1. $\underline{Proof.}$ Assume $g(f(x_0)), g(f(x_1)) \in C.$ Then $g(f(x_0)) = g(f(x_1)) ...
6
votes
3answers
433 views

Can the exponential function be reprsented as infinite product?

Is there any representation of the exponentil function as infinite product (where there is no maximal factor in the series of terms which essentially contributes)? I.e. $$\mathrm ...
1
vote
4answers
652 views

Find zeros of this function:

$$(3\tan(x)+4\cot(x))\cdot\sin(2x)$$ Do I have to multiply them and solve, or one by one, like: $$(3\tan(x)+4\cot(x))=0$$and$$\sin(2x)=0.$$
1
vote
3answers
425 views

Find domain and codomain of this function…

So, the function is: $$ z(x)=\sqrt{-4^x+6\cdot2^x-8}$$ We have to find domain and codomain. There are many more function in the exercise, I just want to know how it's done, so I can do the next ...
2
votes
6answers
1k views

Real world use of even and odd functions

What is the real world use of knowing whether a function is odd or even? Any practical examples? For example, quadratic equations, differential equations and calculus in General is used for, among ...