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

learn more… | top users | synonyms (1)

0
votes
2answers
32 views

Understanding Multiplicities

I am having troubles understanding what 'multiplicities' mean. In example what does $-1/3(multiplicity 2)$ translate into?? To clarify this is for finding zero's in a polynomial function Any help ...
0
votes
1answer
30 views

Graph exponential function

I am having problems understanding why $xe^x + 10e^x$ has two $(x,y)$ intercepts. I understand why there is one $(0,10)$, but am unclear on how to return $(-10,0)$. Any help would be much ...
0
votes
1answer
44 views

help with funky function definition

I've never encountered a function definition like this before and am wondering how you would go from this definition to finding out what features it has (y-intercept, even/oddness, min/max value, ...
1
vote
1answer
118 views

Inverse function of $x+\ln(x)$

How can I find the inverse function of $$f(x)=x+\ln(x).$$ This function has an inverse function (I can prove it) but I couldn't find it. Help please!
2
votes
1answer
128 views

I don't understand the mathematical definition of an inverse function

A function $f:X\rightarrow Y$ is called invertible if there exists a function $g:Y \rightarrow X$ such that: $y=f(x)\Leftrightarrow x = g(y)$ for all $x\in X $ and for all $y \in Y$ In ...
5
votes
2answers
205 views

Help with complicated functional equation

Problem: Let $T=\{(p,q,r)\mid p,q,r \in \mathbb{Z}_{\geq0}\}$. Find all functions $f:T\to \mathbb{R}$ such that: $$f(p,q,r)=\\ =\begin{cases} 0, & \text{ if } pqr = 0 \\ 1 + ...
1
vote
0answers
37 views

How to establish the equivalence of these two statements about integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a, b]$. Then we have $$ \int_{ka}^{kb} s\left(\frac{x}{k}\right) \ dx = k \int_a^b s(x) \ dx $$ for every $k ...
0
votes
1answer
43 views

How to establish this equivalence for integrals of step functions?

First Statement: Let $s$ be an arbitrary step function defined on the closed interval $[a,b]$. Then we have $$\int_{a}^{b} s(x) \ dx = \int_{a+c}^{b+c} s(x-c) \ dx.$$ Second Statement: Let $s$ be ...
0
votes
1answer
17 views

Hypergeometric Distribution Function?

I'm looking for a function that I can use in excel to calculate the probabilities of having certain cards in an opening hand. For example a function that will calculate the probability to get AT ...
0
votes
1answer
64 views

Function with single formula that have non zero value only in one small range

Is it possible to have one formula for a function $f(x)$ that will look like on this picture: Without cases for $x < x_0$, $x >= x_0\space and\space x <= x_1$,$x > x_1$. The formula ...
-1
votes
1answer
56 views

Why does $f(x) = x \cos 4x$ lie below the $x$-axis in ther interval $[\pi/4, \pi/3]$?

How can I explain why function $f(x)$ lies below the $x$-axis in the interval $\left[\dfrac{\pi} 4, \dfrac{\pi} 3\right]$ where $f(x) = x \cos 4x$?
0
votes
1answer
28 views

Graphing functions

I am having problems understanding how to graph the product $fg$ when $f(x) = x$ and $g(x) = |x|$. Any help would be much appreciated!
1
vote
1answer
43 views

I need help proving this theorem (composition of functions)

This is the statement: If $f$ and $g$ are functions, the composition $g\circ f$ is a function with $$D(g\circ f)=\{x\in D(f):f(x)\in D(g)\}$$ $$R(g\circ f)=\{g(f(x)):x\in D(g\circ f)\}$$ The ...
2
votes
1answer
88 views

Is $\mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x)$ onto?

Is $\mathbb{N}\rightarrow \mathbb{N\times N},f(x)=(x,x)$ onto? I am not sure how to tell. Say $b\in N\times N$ this means the codomain is all the different combinations of the natural numbers. But ...
2
votes
2answers
75 views

Concerning Rules of Exponents & Absolute Value

I understand that one of the accepted definitions of the absolute value function is $\left| x \right| = \sqrt{x^2}$. However, I do not understand why if I substitute $-5$ in for $x$ that I can't do ...
3
votes
3answers
170 views

Finding if a function is onto?

Is the following function onto? It is a piece-wise function. Let the function $f:\mathbb{R}\rightarrow \mathbb{R}$ be $f(x)= \begin{cases} 2-x &, x\le 1 \\ \frac{1}{x} &, x>1 ...
2
votes
1answer
42 views

Greatest value of f

If $f'(x)=6-x$ then which of the following has the greatest value? $f(2.01)-f(2)$ $f(3.01)-f(3)$ $f(4.01)-f(4)$ $f(5.01)-f(5)$ $f(6.01)-f(6)$ I know the answer is $f(2.01)-f(2)$ but how to prove?
1
vote
1answer
40 views

Composition of functions which is one-to-one.

$f:Y\rightarrow Z$ and $g:X\rightarrow Y$ If $f\circ g$ is one-to-one then which of the following must be true? 1.$g\circ f$ is one-to-one. 2.g is one-to-one. 3.f is one-to-one. 4.g is onto.
1
vote
1answer
43 views

Find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$

How can I find the minimum value of $\sqrt{x^2+8x+85}+\sqrt{x^2-8x+113}$. I've tried derivating it but didn't reach any result.
0
votes
2answers
85 views

Gradients and functions on matrices

Given a twice differentiable $f: \Bbb R \to \Bbb R$, with continuous second order derivative. We define $$F(x) = \sum_{i=1}^{m}f(x_i)$$ and $$L(x) = \sum_{i=1}^{m}f( \langle a_i, x \rangle+ b_i),$$ ...
0
votes
0answers
48 views

Generalizations of functions

I'm trying to collection examples of mathematical entities that are generalizations of functions. The use of the word "generalization" here doesn't need to be strict, as in every function is an $X$ ...
2
votes
3answers
260 views

How to find know if function is onto?

How do you figue out whether this function is onto? $\mathbb{Z}_3\rightarrow \mathbb{Z}_6:f(x)=2x$ Onto is of course is for all the element b in the codomain there exist an element a in the domain ...
1
vote
3answers
119 views

Is $4x^2-4x+2$ surjective?

Determine whether the function $f_4:\mathbb{R^+}\rightarrow \{x \in \mathbb{R^+} x \ge 1\}$ given by $f_4(x)=4x^2-4x+2$ is injective, surjective or bijective. I will just show parts of the solution I ...
1
vote
3answers
117 views

Why isn't an injection an iif?

Suppose that $f:X \rightarrow Y$ is a function. Then an injection can be defined as: $\forall x_1,x_2 \in X, f(x_1) = f(x_2) \Rightarrow x_1=x_2$ Why isn't it defined instead as follows: $\forall ...
0
votes
0answers
56 views

Reversible smoothing of a two dimensional function (or an image)

Smoothing of an image, or a two dimensional function is quite easy, there are many methods to achieve it, using average of near elements. But how to make it reversible? Maybe DCT (discrete cosine ...
-1
votes
1answer
23 views

Rewriting a quadratic function

i have to find domain of this function $f(x) = \log(10+3x-x^2)$ can i rewrite this as $\log(x^2-3x-10)$? I found the domain but is not the same for 2 forms of function.
1
vote
2answers
86 views

Problem Involving a Generalized Cartesian Product

Let $I$ be a set, and for each $i \in I$, let $U_i$ and $V_i$ be sets. Furthermore, suppose for each $i \in I$, there is a bijection $f_i:U_i \to V_i$. Prove that there is a bijection $g:\prod_{i \in ...
1
vote
1answer
648 views

Need function for 2D sigmoid-shaped monotonic Surface

I am looking for a 2D function, $f(x, y)$ which increases monotonically over the range $(0,0)$ to $(1,1)$. In other words, it will be $0$ at $(0,0)$ and $1$ at $(1,1)$. It will also evaluate to $0$ ...
0
votes
3answers
2k views

What would make a function reflexive, transitive, and/or symmetric?

A binary relation $R$ is a subset of the Cartesian product between two sets $X$ and $Y$, containing a set of ordered pairs $\{(x,y) : x \in X, y \in Y\}$. $R$ is a function if each element of $X$ is ...
0
votes
4answers
41 views

Showing that $f:\mathbb{Z}_4\rightarrow \mathbb{Z}_2$ given by $f(x)=[3x]$ is function?

How can I show this is a function? $f:\mathbb{Z}_4\rightarrow \mathbb{Z}_2$ given by $f([x]_4)=[3x]_2$ where $[x]_n$ is the equivalence class of $x\mod n$. I think it is a function because I cannot ...
4
votes
2answers
202 views

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. [duplicate]

Suppose $f: X \rightarrow Y$, and is one-to-one, and let $A \subseteq X$, prove that $f^{-1}[f[A]] = A$. EDIT: Actually, this identity should hold even if $f$ is not one-to-one (injective), right? ...
6
votes
1answer
77 views

What am I doing wrong in this algebra excercise?

This is my first question here, so please forgive me if the format etc. are not quite right. I've been attacking an algebra question, and my workings are below. There's a mistake somewhere (I don't ...
2
votes
3answers
230 views

How find this P(x) if $ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) $

Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that $$ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) $$ This problem is IMO Shortlist ...
0
votes
1answer
67 views

Derivation of Dirac-Delta with complicated argument $\delta(f(x))$

Recently I learned how to deal with the Derivative of a shifted Dirac-delta. Now I want to go a step further, but are not sure about the solution. Is there a simple way to rewrite terms like this ...
-2
votes
4answers
170 views

Solving a problem using the definition of limit [closed]

How can I solve this using the definition of limit? Prove using the definition of limit that: $$\lim_{x\to 1} (x²-4x)=-3$$ How can I approach this? EDIT: OH my god! Thanks @adam! Maybe you ...
0
votes
0answers
53 views

Sketching the graph of a function with three real roots

I need to solve the following question: Sketch a graph of a function $f(x)$, continuous in all $x \in \Bbb R$, knowing that $f$ has three real roots, that $\lim_{x\to+\infty} \left[f(x)-\frac ...
2
votes
3answers
48 views

Is the inverse of this function unique

Let $f$ be a function from any set(Say $K$) to any set (say $P$) Now: $f(x)=2x+1$ My question:Is it necessary that the inverse of the function is $\frac{x-1}{2}$? This is a problem given in my ...
0
votes
1answer
34 views

Constructing the graph of a function

I need to solve the following problem: Consider the function $g(x) = \ln(x^{2}) + 2$. Construct the functions graph $f(x)=\int g(x)\:\mathrm{d}x$ considering the integration constant equal to ...
0
votes
0answers
21 views

Property of a function.

If $f\colon \mathbb{N} \times \mathbb{N} \to \mathbb{N}$ s.t $f(1,n)=n$ and $f(m,n)\geq f(m-1,n^2+n)$. For such a function,is the following true $x \geq y \implies f(m,x) \geq f(m,y) \ \forall m\in ...
1
vote
2answers
70 views

Prove that addition of a constant on vector spaces is bijective

What would be a nice way to deduce from the vector space axioms that $f : V_1 \longrightarrow V_2, \, x\mapsto x+v$ with constant $v$ is bijective?
0
votes
1answer
45 views

Surjectivity of composition

I know that this question has been posted few times, but I want to check MY proof, because this is my first time trying to prove anything in mathematics. (I'm afraid if I just copy paste their proofs ...
0
votes
1answer
172 views

Understanding the difference between relations and functions.

$R=\{(1,2),(1,3)\}$ is a relation but not function. The logic for this is that if the first element of every ordered pair must remain different, then it is said to be function. Otherwise, it's just ...
0
votes
1answer
27 views

Question about strongly convexity and affinity

For a function $f$, it is said to be strongly convex if for all $x,y$ \begin{equation} (\nabla f(x) - \nabla f(y) )^T (x-y) \ge m \|x-y\|_2^2 \end{equation} for a constant $m \ge 0$ Is it called ...
1
vote
3answers
316 views

Expressing the probability density function of $Ax$ in terms of the pdf of $x$

I understand that, for example, you might have a density function which measures the probability of observing an outcome in a certain interval measured in feet, but someone wishes to use meters ...
0
votes
1answer
42 views

Logarithm with variable base

I am trying to define a function that maps polynomials in the form of $x^{3^n}$ to the value of $n$ in the polynomial, where $n\in{Z}$.* Is is valid to define this function as $log_{x^3}(u)$, where ...
3
votes
2answers
75 views

Function for this game movement graph?

This may be too easy for you, but here goes: I'm creating a kind of ski slalom game where I want the horizontal speed/direction to follow the attached graph. X is time, Y is horizontal speed. Positive ...
1
vote
3answers
185 views

Example of a bijection from the set of real numbers to a subset of irrationals

I need an example of a bijection from the set of real numbers to a subset of the irrationals. I tried something like $f(x)=x+\sqrt{2}$, but where should I map $-\sqrt{2}$?
0
votes
3answers
64 views

Rationals over an interval

Suppose $I$ is an interval $[a,b]$. It is noted that $a$ and $b$ are real integers. Divide the interval into $n$ parts with step size $h=(b-a)/n$. Clearly all the points $a$, $a+h$, ...
0
votes
1answer
24 views

Analyzing a particular type of functions

Let $f$ be a function from $\mathbb{Z}$ to $ \mathbb{Z}$ Now $f(x)=x$ Question: Is $f$ continuous in its domain?( perhaps yes by epsilon delta argument but I don't know if I am justified in doing ...
0
votes
2answers
37 views

maps from infinite sets to infinite sets

I know that the set of irrationals is uncountable, but I feel that there can always a bijection from one infinite set to a subset of another infinite set. Does this sound right? Say, from Irrationals ...