Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
34,601 questions
0
votes
0
answers
58
views
Find a function $f$ which is continuous on $[0, 1]$ and satisfies given conditions
Q) Suppose $0 < a < 1,$ but that $a$ is not equal to $1/n$ for any natural number n. Find a function $f$ which is continuous on $[0, 1]$ and which satisfies $f(0) = f(1),$ but which does not ...
- 1
1
vote
0
answers
52
views
A weaker condition than "equivariancy" getting image sets equality
Let $(G,*)$ be a group. Then, $G\hookrightarrow S_G$ via $\lambda\colon a\mapsto(x\mapsto a*x)$. If $f\colon G\to G$ is a bijection, then again $G\hookrightarrow S_G$ via $\lambda_f\colon a\mapsto(x\...
- 5,402
4
votes
6
answers
194
views
find $\lim_{x\to 0} \frac{\cos(mx)-\cos(nx)}{x^{2}}=\frac{n^{2}-m^{2}}{2} $
I want to find the solution to this limit
Here they give a clue:
https://artofproblemsolving.com/community/c7h500368p2811532
something like this:
$$ \color{green}{y=2x} $$
$$ \color{green}{\lim_{x\to ...
0
votes
0
answers
81
views
Why the area function of $\frac{1}{x^2}$ isn’t the same as its antiderivative?
From what I've seen so far, the area function $A(x)$ of $f(x)$ is some antiderivative of $f$ such that $A(x) = \int_{a}^{x}f(t)dt$ and $A(x) = F(x) + C$. However, when I computed the area function for ...
- 11
1
vote
1
answer
96
views
Determine the smallest possible value of the natural number $ a_1$
Determine the smallest possible value of the natural number $ a_1$, knowing that there exist natural numbers
$ a_1 \geq a_2 \geq \ldots \geq a_{100} \geq 2 $ with the property that
$$
\left\{ \sum_{k=...
- 427
0
votes
1
answer
63
views
How to approach finding many-one and one-one functions using graphical method?
$$f(x) = \frac{x^2 - x + 12}{ x^2 + x + 8}$$
Find whether it's a many-one or one-one function?
I differentiated $f(x)$ using $d(u/v)$ rule and I got:
$$\frac{2(x^2 - 4x - 10)}{(x^2 + x + 8)^2}$$
I ...
- 11
2
votes
2
answers
112
views
How to prove function transformation rules?
I'm a high school student trying to understand function transformations deeply, not just as memorized rules.
Most textbooks say that when we reflect a graph over the $y$-axis — that is, transform $ y =...
1
vote
0
answers
67
views
What are the "boundary conditions" in a functional equation?
Context
This is a purely theoretical question of curiosity, there is not much context.
Let me start by saying that I have little to do with functional equations, but I know that they are equations in ...
- 5,687
40
votes
9
answers
4k
views
Why do we consider there to be gaps between rational numbers, and not between real numbers?
As you may have seen from some of my other questions, I have little knowledge of higher mathematics, and as of now I am in Algebra 2. However, I was arguing with my dad over something, and we stumbled ...
- 1,165
6
votes
5
answers
1k
views
Is the number system for x assumed beforehand when proving the quadratic formula?
When proving the quadratic formula (or any other mathematical equation, definition, formula, etc., from like all the way from basic math to advanced calculus), do we have to assume/declare the number ...
2
votes
1
answer
166
views
Show that a function satisfying $f(xy) = f(ax) + f(by)$ is injective.
$
; \text{Let } a, b \in \mathbb{R}^{*}_{+} \text{ be constants, and:}
$
$
f : \mathbb{R}^{*}_{+} \to \mathbb{R} \text{ a function that simultaneously verifies:}
$
$
(i)\quad f(xy) = f(ax) + f(by), \...
- 427
2
votes
1
answer
645
views
How to rigorously justify plugging values into general physics/probability formulas?
I’m studying Physics and Probability, among other subjects, and I often see general formulas written in class, such as (Physics example):
$F = ma$
Later, my professor computes a specific case directly ...
- 2,605
-2
votes
0
answers
38
views
Is y=f(-x-h) a vertical reflection followed by a translation, or the other way around?
I am currently working on a math project and have to transform functions to make some sort of art/image. I am confused on the order of transformations in this specific instance, where y=f(-x-3). To ...
- 363
0
votes
0
answers
32
views
Lagrange Multipliers and Lagrange Functionals [duplicate]
I am a high school student (Please excuse my non rigorous terminology) and for a mathematics paper I am proving that for any closed curve with a fixed perimeter that doesn't intersect itself, the ...
- 21
0
votes
0
answers
76
views
$f(x)$, $g(x)$ and $h(x)$ are assumed to be integrable, is there a simpler set of assumptions I can use instead?
Background
I'm working on formalising a theorem in lean.
...
- 1,050