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

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0
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1answer
54 views

How to find the domains of functions $f(x) = x-5$, $g(x) = \sqrt{x-5}$, and of their sim?

I've been studying on Study Plan Practice, on MyMathLab for my College Algebra class. We're going over the Algebra of Functions right now and several things don't make much sense. The question is: ...
1
vote
1answer
30 views

Problem on function definition

I'm trying to solve this problem about functions: "Explain why $x^2-4=0$ is not a real function of real variable." I have yet solved many similar problems, but now i have a doubt; is my solution ...
0
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4answers
49 views

Find the domain of the function $f(x) =\frac{x+4}{x^2-9}$

I need to find the domain of the function $\;f(x) =\dfrac{x+4}{x^2-9}.$ My answer was: $(-\infty, -3)\cup(3, \infty)$. The book's answer was: $(-\infty, -3)\cup(-3,-3)\cup(3,+\infty)$ It's question ...
4
votes
1answer
220 views

Proving a function is increasing

Is there a nice way to prove that $f(x)=x^3+x^2+x-3$ is strictly increasing without making use of derivatives or any other advanced concepts ? I'm trying to explain it to a 9th grader, but I can't ...
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0answers
7 views

Calculating Bloch-Wigner dilogarithm

Is there some tool/calculator (or some tables) for explicitly calculating values of the Bloch-Wigner dilogarithm?
1
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2answers
67 views

$\left|f'(x)\right| \leqslant g'(x)$ implies $\left|f(x)-f(a)\right| \leqslant \left|g(x)-g(a)\right|$

Suppose that both $f$ and $g$ are real, differentiable functions over $[a, +\infty)$, where $a$ is a real number; and that for all $x \geqslant a$, $\left|f'(x)\right| \leqslant g'(x)$. Prove that ...
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1answer
20 views

Proving $A = f^{-1}(f(A))$, given an injective function $f : X \to Y$ with $A \subset X$

Let $f : X \to Y$ be a one-to-one function with $A \subset X$. Show that $A=f^{-1}(f(A))$. My try: If $x \in A$, then $f(x)\in f(A)$. By definition, $f^{-1}(f(A))=\{x\in A : f(x) \in f(A)\}$. ...
4
votes
3answers
139 views

What is the order of operations in trig functions?

Is $\sin(x)^2$ the same as $\sin^2(x)$ or $\sin(x^2)$? I thought it would mean the former interpretation, $\sin^2(x)$, rather than the latter, but my teacher and I had a long argument on this and in ...
0
votes
2answers
50 views

Non-bijective isomorphism in a category of of sets.

I have been commanded on homework to find a non-bijective isomorphism in a category whose objects are sets, whose morphisms are set maps, and composition is the usual function composition. So our ...
1
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0answers
22 views

operator vs operation vs function vs procedure vs algorithm

I have a vague understanding of what operator, operation, function, procedure, algorithm mean in general. I am heavily biased towards computer science. Do you agree with them? What are the generally ...
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1answer
45 views

What is the cardinality?

Let $A=\left\{1,2,\cdots,10\right\}$ Let $f,g:A\to A$. Consider the equivalence relation $$ fRg \iff \exists h:A\to A. f=h\circ g$$ where $h$ is invertible. Now, let $g(x)=5$: Why is $\left| ...
2
votes
1answer
26 views

Notion of surjective functions

I have some issues with fully understanding the concept of surjection. For the proof of $A_{m}$ being a countable set for each m $\in$ N, then the union $A = \bigcup_{m=1}^{\infty} A_{m}$ is ...
0
votes
1answer
77 views

$f(x+h)$ not equal to $f(x) +f(h)$???

I'm taking College Algebra at a local community college, and I just wasn't able to follow how my professor came to these conclusions. (3 separate times.) $$\frac{f(x+h) - f(x)}{h},$$ $$f(x) = ...
1
vote
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
votes
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 ...
2
votes
4answers
79 views

Showing Surjectivitity of $f(x) = x^3$

I want to show that the function $f: \mathbb{R} \to \mathbb{R},\; f(x) = x^3$ is surjective. First Question: If a function has an inverse, it is bijective yes? Second Question: Is my process ...
3
votes
2answers
54 views

If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$

Consider the statement: If $f$ takes $[-1,1]$ onto $[-1,1]$ then $f^{-1}(\{f(0)\})=\{0\}$. My book tells me this is suppose to be false, but I don't understand why. We know: If $f:X\to Y$ has ...
1
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3answers
145 views

Is there a function whose derivative is $|x|$?

Is there a function $y=f(x)$ such that $$\frac{df}{dx}|_{x=a} =|a|$$ for all $a\in \mathbb R$? I'm in a debate with my friend over it and we are stuck
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0answers
11 views

evaluate difference quotient of function

x+9/x+2 F(x)-F(5)/x-h evaluate the difference quotient for the given function. Simplify your answer. I got as an answer of 1/x-5 which i do not think is correct. what am i doing wrong?
0
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1answer
30 views

Notation Abuse for dependent variable differentiation

Let $z(x,y,t) = x^2+y+t^2$ where $x(t)=t$. $\left(\dfrac{\partial z}{\partial x}\right)_{y,t} = 2x$ If you substitute in x for a t, you get the following. $z(x,y,t) = x^2+y+xt$ ...
1
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3answers
37 views

Differentiating a function by simplification.

If we consider a function: $f\left(x\right)=\dfrac{x-1}{2x^2-7x+5}$ This function is not defined at x=1 and x=5/2. So if we differentiate this function by u/v method we have: ...
3
votes
3answers
508 views

How many functions are transitive?

Let the set of all functions defined as: $\left\{a,b,c,d\right\} \rightarrow \{a,b,c,d\}$ How many functions are transitive? I've been told to use the fact that a function is transitive iff "it's ...
1
vote
1answer
26 views

Continuous functions, its inverse (if exists) and intersections graphically

I have a question regarding graphical intersection between a continuous function and its inverse (if exists). Suppose $f$ is a real continuous function and $f^{-1}$ exists. Can anyone assist in ...
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0answers
19 views

Name and properties of this “triangular” function

When doing calculations in Quantum Field Theory, a function $$f(a,b,c) = a^2 + b^2 + c^2 - 2ab - 2bc - 2ca$$ keeps popping up. I've seen somewhere that it's called the triangular function, but I can't ...
0
votes
1answer
16 views

Number of variables and dimension of a function

Why is a function $f(x)$ called a single-variable function if it has coordinates represented by $x$ and $y$? Can it be called a 1D function if its plot is 2D? Subsequently, can two-variable functions ...
3
votes
2answers
61 views

In composition of two mappings, can the outer mapping access the arguments of the inner mapping?

In composition of two mappings, can the outer mapping access the arguments of the inner mapping? Here is an example to illustrate my question and my thought. E.g. $f: \cup_{n \in \mathbb N} \mathbb ...
1
vote
1answer
17 views

Finding the limit of a function of 2 variables

Hello I am having a hard time understanding this assignment, I need to find the limit for the equation below as it approaches $(0,0)$, but I don't know how to do that, any help is greatly appreciated. ...
4
votes
2answers
69 views

How many numbers are less than million such that their digits sum is $\le 19$?

How many numbers are less than million such that their digits sum is $\le 19$? This question is a Generating-Functions exercise. The solution claims the answer is the coefficient of $x^{19}$ ...
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2answers
45 views

Help explain proof that $\mathbb{Q}$ is denumerable

A set $X$ is said to be denumerable if there is a bijection $\mathbb{Z}^+\rightarrow X$ Statement to prove: The of rationals $\mathbb{Q}$ is denumerable. Proof: Define $f:\mathbb{Q} ...
4
votes
1answer
41 views

Is it of any value to express a function as the sum of an even and an odd function?

So I learned about this formula $$ f(x) = \frac{f(x)+f(-x)}{2}+\frac{f(x)-f(-x)}{2} $$ and I'm wondering, is it of any value to express a function in this form?
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2answers
23 views

How to prove that this function of two variables is a surjection?

Let: $A=\{a_1,a_2,\dots,a_m, \dots\}$ $B=\{b_1,b_2,\dots,b_n,\dots\}$ Define $f:A\times B \rightarrow \mathbb{Z}^+$ as: $(a_m,b_n) \mapsto \frac{1}{2}(m+n-2)(m+n-1)+n$ How do I show that $f$ is ...
1
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0answers
22 views

Representing a function by simplification.

The concept of 'functions and limits' always seemed confusing to me. I came across this question which occurred to me as a 'basic' of limits. Q. Check whether the following functions are the same: ...
2
votes
1answer
52 views

A game involving Even and Odd functions

I was doodling around today and thought of this fun game. Two players take alternate turns playing this game. A function from now on refers to real valued functions with domain $\mathbb R$, and Odd ...
1
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2answers
50 views

Why does the graph not show the vertical asymptote?

I have a rational function in which the denominator is equal to zero when $x = -1.4142135$. So: $f(x)=\dfrac{1}{x + 1.4142135}$ Then the vertical asymptote at $x = -1.4142135$ is pretty clear. But ...
1
vote
1answer
18 views

Creating a scoring algorithm for voting app

I have just created what I thought to be a good scoring algorithm until I looked at some demo results. It now looks pretty floored though I have a table with a number of images I have another table ...
2
votes
1answer
27 views

Solving inequation where one of the terms is a log

I trying to find the value for which $n^2 -n +1$ is less than $ 6n\log_{2}{n} +2n $ where n is a power of $2$. Trying it iteratively using a CAS you find that $n = 64$. How can $n^2 -n +1 < ...
1
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1answer
26 views

Help to prove this inductively defined function is surjective

Suppose that $A$ is a infinite subset of $\mathbb{Z}^+$. We construct a bijection $f:\mathbb{Z}^+ \rightarrow A$ and define $f(n)$ inductively as follows: Base case: Let $f(1)$ be the least element ...
0
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0answers
12 views

From essential oscillation to a continuous representative

Let $u$ be a measurable function such that for every(former: a.e.) $x\in \Omega$ there holds for sequences $R_n,\delta_n\to 0$ that$$\omega_n:=ess-osc_{B_{R_n}(x)} u\leq \delta_n. \tag{1}$$ Edit: ...
-1
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0answers
30 views

Is there a function that will produce unique output based on two integers? [duplicate]

Presume two integers A and B are obtained. These integers may follow any rules as to their generation (i.e., if primes would be useful, then let them be primes) but cannot influence each other's ...
2
votes
1answer
43 views

Derivative of an Inverse Function

Can someone please give me a simple proof of this- If $f$ is differentiable on an interval containing $c$ and $f'(c) \neq 0$, then $f^{-1}$ (inverse of $f$) is differentiable at $f(c)$. I can see ...
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0answers
58 views

A question about $f(f(x))=x^2+x$ [duplicate]

If $f(f(x))=x^2+x$, then what is $f(x)$? I have already Taylor's expansion and power series.
18
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4answers
3k views

Why is an image called an “image”?

Given a function $f : A \to B$, the image, denoted by $\operatorname{Im}f$ is the set of all $f(x)$ where $x \in A$. Why do we call this set the image? When was it first used, and what motivated its ...
2
votes
2answers
38 views

Question on Partial Derviatives

For function $f(x,y) = x^2 y$ The partial derivatives for $x$ is $2.x.y $. I'm new to such math equation and i'm learning them now. May i know why is it so? Thanks!
2
votes
2answers
51 views

Finding the limit of $F(x)=\frac{x^2-4}{|x+2|}$

Let $F(x)=\dfrac{x^2-4}{|x+2|}$ and find the following limits $(a) \; \; \lim_{x \to -2^-}F(x)=$ $(b) \; \; \lim_{x \to -2^+}F(x)=-4$ $(c) \; \; \lim_{x \to -2}F(x)=DNE$ I substituted $-2$ to find ...
0
votes
1answer
29 views

Find a function that maps x,y to $[0, n ( n + 1) / 2)$

Can you find me a bijective function that maps positive integers $x, y$ such that $0 \leq x < y \leq n$ to integers in $[0, n(n+1)/2)$ to use as a hash function?
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2answers
15 views

Prove if $g \circ f$ is $1-1$ and $f$ is onto, show that $g$ is $1-1$

Let $f: A \rightarrow B$ and $g: B \rightarrow C$. $g \circ f: A \rightarrow C$. But where do I use the fact that $f$ is onto?
0
votes
1answer
16 views

Domain of a multiple logarithmic function.

Find the domain of the following function: $f\left(x\right)=log_4\left(log_5\left(log_3\left(18x-x^2-77\right)\right)\right)$ My text provides a solution which goes like: => ...
0
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0answers
15 views

Check proof of union of denumerable sets is denumerable too

I need to prove: If $A$ and $B$ are denumerable sets then so is their union $A\cup B$. In this case, denumerable is defined as: A set $X$ is said to be denumerable if there is a bijection ...
0
votes
2answers
18 views

functions in form of $f\circ g$

Express the function in the form $f\circ g$. (Use non-identity functions for $f$ and $g$.) $F(x)=(6x+x^2)^4$ I understand you have to find what each $f(x)$ equals and $g(x)$ equals but not really sure ...
0
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3answers
43 views

What would be the inverse function for the following condition?

What would be the inverse function condition for the above question.