Questions tagged [functions]
For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.
34,629 questions
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4
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Find $ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $ [closed]
Problem
$$ \lim_{x\to+\infty} \left( \frac{x^{2}+3}{3x^{2}+1} \right)^{x^{2}}=0 $$
My Work
$$ \lim_{x\to+\infty} \left(\frac{1+\frac{3}{x^{2}}}{1+\frac{1}{3x^{2}}}\cdot\frac{1}{3} \right)^{x^{2}\cdot\...
4
votes
1
answer
128
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Does there exist a function such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$?
Does there exist a function $f: \mathbb{R} \to \mathbb{R}$ such that $f(x), f(x)+\sqrt{3}, \sqrt{2}-f(x), f(x)+x$ are irrational for all irrational $x$?
My attempt: I couldn't come up with any good ...
- 553
0
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0
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predicting the point of changing trend types in tan nesting functions
(from a hsgs high school math group chat)
Let there be function $t(n,k)=q$, such that $k$ is in radians, and that $n$ is how many $\tan$ functions are nested in each other such that
$$t(1,k)=\tan(k)$$
...
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3
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Find $ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $
$$ \lim_{x\to \infty} \left( \frac{x-4}{x+1} \right)^{x+3}=e^{-5} $$
I know that I am not making any change in the expression, I am just re-expressing it
$$ \lim_{x\to \infty} \left( 1+\frac{-5}{x+1} \...
-2
votes
3
answers
104
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Find $ \lim_{x\to\infty} (\frac{3+x^{2}}{4x^{2}-1})^{x^{2}}=0 $ [closed]
I tried but I obtein an indeterminacy. I think that I am ignoring any property of notable limits maybe
$$ \lim_{x\to\infty} (\frac{3+x^{2}}{4x^{2}-1})^{x^{2}}\to $$
$$ \lim_{x\to\infty} (1-\frac{3x^{2}...
2
votes
0
answers
171
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Do I need to explicitly state $x \neq 0$ when an equation already implies that it cannot be zero? For example, $ \frac{1}{x} = x y^2$.
Consider the equation
$$
\frac{1}{x} = x y^2.
$$
Since the expression $\frac{1}{x}$ appears, the equation is only defined when $x \ne 0$.
My question is:
Do I need to explicitly write the domain ...
0
votes
1
answer
62
views
Yet another nice functional inequality [duplicate]
Need to find out all the functions $f \colon \mathbb{R} \to \mathbb{R}$ such that
$$f(x+y) \leq f(xy).$$
Since $f(x) \leq f(0)$ for every $x \in \mathbb{R}$ and $f(0) \leq f(-x^2)$, it follows that $f(...
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parabolic speed of an arrow from a tower
We had an exercise where we needed to calculate a couple of things but here is the question translated:
A guy is standing on a tower at the height of $10$ meters and shoots an arrow. The height can be ...
0
votes
2
answers
88
views
partial derivatives of an unknown function [closed]
I’m trying to understand how to compute the partial derivatives when the function is given implicitly.
For example, suppose we have an equation
$x^2+y^2+z^2=a^2$
where $z$ is considered to be a ...
-1
votes
2
answers
118
views
Find $ \lim_{x\to \infty} \frac{x^{2}+bx+c}{x-n}=\infty $
$$ \lim_{x\to \infty} \frac{x^{2}+bx+c}{x-n}=\infty $$
recalling the trinomial of the form:
$$ x^{2}+bx+c=(x+n)(x+m) $$
for some $n$ and $m$ such that
$$ m+n=c $$ $$ m\cdot n=b $$
$$ \lim_{x\to \infty}...
7
votes
2
answers
352
views
Find all the functions $f \colon \mathbf R \to \mathbf R$, such that $f(x+\frac{1}{y})+f(y+\frac{1}{x})=2f(xy)$ for all $x,y \in \mathbf R$.
the problem
Find all the functions $f \colon \mathbf R \to \mathbf R$, such that $$f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{x}\right)=2f(xy)$$ for all $x,y \in \mathbf R$.
my idea
Plugging in ...
- 413
0
votes
0
answers
60
views
How to prove that $f(\theta)$ reaches its minimum when $w_1w_2=0$?
I have the following function:
$
f(\theta)=\int_{-1}^{1} \max\!\left(0,\; \min\!\left(1,\; \frac{v - w_1 \xi}{w_2}\right) \;-\; \max\!\left(-1,\; \frac{-v - w_1 \xi}{w_2}\right)\right)\, d\xi,
$
where
...
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4
votes
4
answers
258
views
Bounds for $\cos^3(x)+ \sin^3(x)$ on $(0, \pi/4]$
I was wondering how to show that on $(0, π/4], -1 < \cos^3(x)+\sin^3(x) < 1$.
I started by bounding $\cos(x)$, then $\sin(x)$ over this interval, then, exploiting the strict growth of the ...
- 41
5
votes
2
answers
218
views
Prove that a function with given condition is constant
Let $X \subset \mathbb{R}$ be some nonempty subset and let $f : X \to X$ be a function satisfying
$$
f(x) + y \in X \iff x \neq y
$$
for all $x,y \in X$. Prove that $f(x) + x$ is constant over all $x \...
- 117
0
votes
1
answer
65
views
Relative and absolute extrema
I am having some doubts about the definition of relative and absolute extrema, which have arisen from the function $f(x)=\sqrt[3]{(x^2-1)^{2}}$.
For me, this function has:
Relative maximum but not ...
- 599
0
votes
1
answer
63
views
Name for the concept of simplifying functions when a pole is present
I'm trying to find a concrete reference for the rule/concept of simplifying function expressions with a discontinuity present. For instance, $f(x) = x^2/x$ simplifies to $x$, but that doesn't mean ...
- 101
0
votes
1
answer
66
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Infinite Tetrations Iterated Infinitely.
Im an IB AAHL student, and I have been playing around with the fascinating process of infinitely iterated tetrations, instead of studying. I thought of sharing it with other individuals interested in ...
2
votes
2
answers
131
views
Proving that a function is discontinuous everywhere
Let any complex number be represented as $c = a + ib$, where $i^2 = -1$ and $a,b \in \Bbb{R}$.
Step 1. Definition of the domain and function $f$
Define the set $A = \{ a + ib : a,b \in (-1,1) \}$.
...
1
vote
3
answers
104
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Prove that $f(x)=x^4\cos(1/x)+2x^4$ has a strict local minimum in $0$ that is not an isolated minimum
Consider the function defined as $f(x)=x^4\cos(1/x)+2x^4$ for $x\neq 0$ and $f(0)=0$. It is clear that $0$ is a global strict minimum (therefore also local strict) because $\cos x\ge-1$ for every $x\...
- 1,256
1
vote
1
answer
128
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Which approach in $ \lim_{u\to0}\frac{2\sin^{2}(\frac{u}{2})}{\sin(u)} $
I have two approaches but I think only one is correct, I need some clarification
multiply by variable raised to the largest exponent $$ \color{red}{u^{2}} $$
$$
\begin{aligned}
&\lim_{u\to0}\...
0
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0
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68
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Find a function $f$ which is continuous on $[0, 1]$ and satisfies given conditions
Q) Suppose $0 < a < 1,$ but that $a$ is not equal to $1/n$ for any natural number n. Find a function $f$ which is continuous on $[0, 1]$ and which satisfies $f(0) = f(1),$ but which does not ...
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5
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8
answers
371
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find $\lim_{x\to 0} \frac{\cos(mx)-\cos(nx)}{x^{2}}=\frac{n^{2}-m^{2}}{2} $
I want to find the solution to this limit
Here they give a clue:
https://artofproblemsolving.com/community/c7h500368p2811532
something like this:
$$ \color{green}{y=2x} $$
$$ \color{green}{\lim_{x\to ...
1
vote
2
answers
264
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=...
- 413
2
votes
2
answers
136
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 =...
42
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 ...
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