Questions tagged [functions]

For elementary questions about functions, notation, properties, and operations such as function composition. Consider also using the (graphing-functions) tag.

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laplace transform of two different functions

Is it possible for two different functions to have the same laplace transform?.In this sense how do we know that the inverse laplace transform gives exclusively one function?.if it is not possible ...
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Find the sign of $f(x) = \cos(n \pi \sin(x))$ and $g(x) = \sin(n \pi \cos(x))$ with $n \in \mathbb N$

So I want to find the sign of $f(x) = \cos(n \pi \sin(x))$ with $n \in \mathbb N$ and $x$ $\in$ $\mathbb R$ with proof. To be clearer, for which values of $n$ is the function $f(x) \gt 0$ and for ...
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Showing that $f((x_1,x_2),(y_1,y_2)):=x_1-y_1$ is continuous.

Define $f:\mathbb{R}^2\times\mathbb{R}^2\rightarrow\mathbb{R}$ by $$ f((x_1,x_2),(y_1,y_2)):=x_1-y_1 $$ I want to show that $f$ is continuous. I already know that the function $g(x,y):=x-y$ is ...
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How would you name this function

We have the following function. I'm trying to give this a functional name. Specifically what I'm looking for is a name to give $U_o$. $U_o = \frac{(\Sigma (U_i * A_i))}{A}$ for $i = 1, n$ $U$ is a ...
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1 vote
2 answers
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Determine by definition whether the function is Uniform continuity $f(x)=\dfrac{\sin(x)}{x}, x\in(0,\pi)$

Determine by definition whether the function is Uniform continuity $$f(x)=\dfrac{\sin(x)}{x}, x\in(0,\pi)$$ $$|\dfrac{\sin(x)}{x}-\dfrac{\sin(y)}{y}|\leq|\dfrac{y\sin(x)-x\sin(y)}{xy}|\leq ??$$ I ...
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2 votes
1 answer
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How many functions are possible for the given differential equation?

There was this question asked in a competitive examination , the solution of which is very confusing to me. The number of differentiable functions $y:(-\infty, \infty) \rightarrow [0, \infty)$ ...
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How many different X arrays are there without Y consecutive Z's

You are given a positive integer $N$, $k$, $l$ and distinct symbols $x_1, x_2, x_3 ... x_k$. $f(N)$ indicates the number of arrays (consisting of $x_1, x_2, x_3 ..., x_k$) of length $N$ that don't ...
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28 views

Number of differentiable functions satisfying $y' = 2 \sqrt{y}$

The following question was asked in KVPY $2021$ held on $22$nd May $2022$: The number of differentiable functions $y:(-\infty, +\infty) \to [0, \infty)$ satisfying $y' = 2\sqrt{y}$, $y(0) = 0$ is (A) ...
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3 answers
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What is the minimum value of the function $f(x)= \frac{x^2+3x-6}{x^2+3x+6}$?

I was trying to use the differentiation method to find the minimum value of the person but it did not give any result, I mean when I differentiated this function and equated to zero for finding the ...
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6 votes
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$f,g$ convex, increasing real functions with $\frac{f(x)}{g(x)}\to 1.$ Does $\frac{f^{-1}(x)}{g^{-1}(x)}\to 1\ ?$

Let $f,g:\mathbb{R}\to\mathbb{R}$ be convex strictly increasing real functions (so we have both $f(x)\to\infty $ and $g(x)\to\infty$ as $x\to\infty),$ and suppose further that $\frac{f(x)}{g(x)}\to 1.$...
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Doubt on the domain of $f(x)=\int_{1}^{x}\frac{e^t}{t}dt$

Let study the function $f(x)=\int_{1}^{x}\frac{e^t}{t}dt=-\int_{x}^{1}\frac{e^t}{t}dt$. First of all I want to determine the domain. The integrand function is defined for all $t\neq 0$. This means ...
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If $f(n)= n-2$ for $n>3000$ and $f(n)=f(f(n+5))$ for $n\leq 3000$,then find the value of $f(2022)$?

Let $f(n)=n-2$ for $n>3000$ and $f(n)=f(f(n+5))$ for $n\leq 3000$ I have to find $f(2022)$. I need to find out $\underbrace{f(f(f(f(..(f}_{197\text{ times}} (3002)\cdots)$ because when $n>3000$ ...
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Find the maximum of $\sin(A\cdot\cos(x))$ and $\cos(A\cdot\sin(x))$ algebraically

I need to find, algebraically, the maximum of $f(x)=\sin(A\cdot\cos(x))$ and $f(x)=\cos(A\cdot\sin(x))$ with $A\in \mathbb R$ with proof. So how I can find the maximum of these functions without using ...
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Let in domain $G⊂R^2$, the $ f:G→R, f∈C^1(G)$, and $\frac{ ∂f}{∂y}(x,y)≡0$ in G. Is it possible to assert that $f$ does not depend on $G$?

Question : Let in the domain $G\subset \mathbb{R}^2$, the function $f:G\rightarrow R, f\in C^1(G)$, and $\frac{\partial f}{\partial y}(x,y)\equiv0$ in G. Is it possible to assert that the function $f$ ...
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Place a point on a line, given only the two points (not the equation)

Suppose I have this line, but I do not know the equation to draw it. I only know that there are 2 points on this line : point 1 at (1, 2) and point 2 at ...
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How to properly transform functions?

If $f(x) = \lvert x\rvert$, the graph is this: If $f(2x) = \lvert 2x\rvert$, the graph is this (the blue line). This is as expected. Every point was multiplied by a factor of 1/2: If $f(2x+5) = \lvert ...
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0 votes
2 answers
86 views

Domain and range of $f(g(x))$ given only the graphs of $f(x)$ and $g(x)$? [closed]

I am a little confused on how to find the domain and the range of the function $f(g(x))$ when only given the two graphs. I kind of understand the solution, but I am struggling to use the domain of $g(...
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1 answer
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Are there nonlinear differentiable functions that are positively homogeneous of order 1?

A function $f:\mathbb{R}\mapsto \mathbb{R}$ is positively homogeneous of order 1 if $f(tx) = tf(x) \quad \forall t>0$. For instance, $f_{\alpha}(x) = \alpha x$ is a positively homogeneous funnction ...
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Notation: function returning the element of a partition containing $x$

Suppose $Y=\{y_1,\ldots,y_m\}$ partitions the set $X=\{x_1,\ldots,x_n\}$. I would like to define a function $y: X \to Y$ which returns $y \in Y$ if and only if $x \in y$. Is there a way to write this ...
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1 vote
0 answers
57 views

Find $gf(x)$ in terms of $h$ and $k$ for the functions $f(x) = 5x - 1$, $g(x) = hx + k$.

Find $gf(x)$ in terms of $h$ and $k$. \begin{align*} f(x) & = 5x - 1\\ g(x) & = hx + 2k \end{align*} What I've tried: \begin{align*} g(5x-1) & = h(5x-1)+2k\\ & = 5hx-h+2k \end{...
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2 answers
52 views

Characterizations for normal distribution and Poisson distribution

Let $X$ be a real valued random variable such that for all $f \in C_c^{ \infty}( \mathbb R)$ we have $ \mathsf E(Xf(X))= \mathsf E(f'(X))$. Show that $X$ has the standard normal distribution. Let $ \...
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12 votes
1 answer
109 views

Why does it seem more natural to think of $r$ as a function of $\theta$, rather than the other way around?

When teaching functions in polar coordinates, the nearly universal practice is to consider functions of the form $r = f(\theta)$. I think I have never seen any examples in which $\theta$ is expressed ...
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Is this part of the Banach density function $BD$ subadditive?

Let $A$ be a set and consider the function $$f(n) = \max_{x \in \mathbb{Z}} |A \cap [x+1,x+n]|,$$ where $|\cdot|$ denotes cardinality and $n \in \mathbb{N}$. I am reading a paper "An elementary ...
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0 answers
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How to construct a function to find the first member of an ordered pair given the pair's *n*-th position in a zig-zag enumeration

i have just begun my journey to teach myself mathematics, and of course, that means being stuck very often and incapable of seeing a way out or around even simple problems. I am currently slowly ...
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0 answers
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Relations properties, powerset

If $X$ is the powerset of $\{5,6,7\}$ and $Y$ is a relation on $P$ such that $Y = \{(n,m) | n \text{ and } m \text{ have the same smallest element}\}$. Would $Y$ be reflexive? Symmetric? Anti-...
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1 vote
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Finite union of epigraphs

In page 6 of H.H. Bauschke and P.L. Combettes, Convex Analysis and Monotone operator Theory in Hilbert spaces, there is a small lemma saying the following: Let $(f_i)_{i \in I}$ be a family of ...
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-3 votes
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How many functions respect these two properties?

Let $A=\{0,1,2,.....,8\}$, $B=\{0,1,2,.....,11\}$ Let $f:A↪B$ a function defined $\forall x,y \in A$ I define two properties: $f(x+y\mod9)=f(x)+f(y)\mod12$ $f(xy \mod9)=f(x)f(y)\mod12$ How many ...
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2 votes
0 answers
52 views

Help showing that two sets are equal.

Let $A,B\in\mathbb{N}$ and define $$\tag{1} U(A, B):=\{K \subseteq \mathbb{N} \mid A \subseteq K \subseteq \mathbb{N} \backslash B\} \subseteq 2^{\mathbb{N}} $$ Define further $f: 2^{\mathbb{N}} \...
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  • 487
2 votes
1 answer
48 views

Is there a function $f$ from reals to reals such that every non-vertical line intersects $f$ infinitely many times?

Does there exist a function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for every non-vertical line $L$ in $\mathbb{R}^2$, $L$ intersects the graph of $f$ infinitely many times?
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1 vote
1 answer
43 views

Number of functions under some conditions

let $A=\{0,1,2,\dots,8\}$ and $B=\{0,1,2,\dots\}$. How many function from $A$ to $B$ can be defined such that the following will be hold: $f(x+y \bmod{9}) = f(x)+f(y) \bmod{12}$ $f(x \cdot y \bmod{9}...
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1 vote
0 answers
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Is there a simpler function $f$ equivalent to $f(x)=af(x-1)^2+bf(x-1)+c$?

I have been using the function $$f(x)=af(x-1)^2+bf(x-1)+c$$ for a project, but wanted to know if there was a closed form of the equation or a form of the function in relation to $f(0)$ or $f(1)$. If ...
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1 vote
1 answer
36 views

Are functions defined or modeled as sets?

In his book Analysis I, Tao says functions are not technically sets. In this post, all the answers agree that functions are not sets, especially the one given by Peter Smith. But almost every book on ...
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1 vote
2 answers
62 views

Find the minimum of $f(x)=x^2-x+1+\sqrt{2x^4-18x^2+12x+68}$.

WA gives the result $9$. But how to solve it by applying inequalites?
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Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$?

Question: Let $q\in\mathbb{Z} $. What is the smallest period of $\sin(2t)−\sin(qt)$? My attempt: the period of $\sin(t)$ is $2\pi$, so the period of $\sin(2t)$ is $\pi$ and the period of $\sin(qt)$ is ...
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  • 153
1 vote
1 answer
29 views

Equinumerous finite sets

Let $A$ and $B$ be two finite sets and we have two functions $f: A\to B$ and $g: B \to A$ such that for any $a\in A$, $g(f(a))=a$ for any $b\in B$, $f(g(b))=b$. Then can I conclude that $A$ and $B$ ...
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1 vote
1 answer
38 views

Does a vertical asymptote mean it is not a function?

I know that a function is defined as something where very input has a unique output, but does "undefined" or "infinity" count as a unique output? In other words, would a reciprocal ...
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0 answers
84 views

Is there a "canonical" way to "flatten" a path function to $f(t)=(x(t),y(t))$ (or otherwise traverseable) coordinates with injective $f$?

Is there a "canonical" way to "flatten" a path function to $f(t)=(x(t),y(t))$ (or otherwise traverseable) coordinates with injective $f$? That is, since the path function does not ...
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3 votes
2 answers
46 views

Understanding random variables as functions

First of all, I have read What is a function and I have understood it basically and it is clear to me that in order to caluclate statistics "things" have to be transformed or mapped to ...
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0 votes
1 answer
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Method for getting the real roots of exponential and logarithmic type equations

Lets real roots of equation be $x_1,x_2..$ of $\log_{2021} x = 2022-x$ and $2021^y = 2022-y$ be $y_1 ,y_2..$ find $x_1+y_1+x_2+y_2..$. My method from graph of all three functions separately we can ...
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0 votes
0 answers
26 views

How to know if the function is convex

How I can know if this function is a convex and why ? My understanding is the exp is convex and if we add positive number to a convex we return a convex then the log for convex sometimes return convex ...
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  • 1
0 votes
1 answer
39 views

The largest $f(x)$ for which $f(x)/g(x)\to 0$ as $x \to \infty$ [closed]

For a generic (real) function $g(x)$ and $x \in \mathbb{R}_{\geq 0}$, how can I find the largest $f(x)$ for which $f(x)/g(x)\to 0$ as $x \to \infty$? This question is not for any reason beyond ...
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0 votes
1 answer
26 views

what does *The range of k that the number of real solutions is maximum* mean in this context?

For a constant $k$, we consider the number of distinct real solutions of equation $x|x^2-3x+2|=k$. The range of k that the number of real solutions is maximum is $? < k < ?$, and the maximum ...
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-1 votes
0 answers
41 views

Can we solve this differential equation or there is some other way [closed]

$f(x)$ is defined for $x≥0$ and has a continuous derivative. It satisfies $f(0)=1,f'(0)=0$ and $$(1+f(x))f''(x)=1+x$$ Then which of the following is not a possible value of $f(1)$? $2$ $1.75$ $1.5$ $1....
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1 vote
1 answer
40 views

Numerical Differentiation Table

The following data was collected by measuring the distances in kilometres that a moving object travels over time (t) in seconds t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 s 0.0 9.0 20.0 34.0 48.0 64.0 80....
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-1 votes
0 answers
21 views

f(x) is defined for the interval [a, b] and continuous, where f(a) = f(b) = 5 and b - a = 24, show that there exist x such that f(x) = f(x + c). [duplicate]

Is this possible for any c, 0 < c < 24? If not, then in which case is it possible and how to show either of the cases with Intermediate Value Theorem?
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  • 1
0 votes
1 answer
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f(x) is defined for the interval [a, b] and continuous, where f(a) = f(b) = 5 and b - a = 24, show that there exist x such that f(x) = f(x + 12). [duplicate]

I know that this question can be solved using Intermediate Value Theorem, but I don't know how to show that such a point exist between the interval. Ideally, it would be very helpful to show that ...
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  • 1
2 votes
2 answers
66 views

Multiplying by $1$ adds a solution to an equation

I have a question, which is motivated by my book's solution to finding the inverse function of $f(x)=\frac{x}{1-x^2}$ with the domain of $f(x)$ restricted the open interval $(-1,1)$. The questions are ...
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  • 91
-1 votes
1 answer
39 views

How to define the zero function

I stumbled upon this and right at the start got confused by what the OP meant by $f(x) = -f(x)$ being the "zero function" $f(x) = 0$. Can someone walk me through the what and how of why this ...
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1 vote
1 answer
54 views

Determine if the functions are differentiable at $x=4$

analyze the function $$f(x)= \begin{cases}\sqrt{4x}+11 & \text{if }x \ge 4 \\ \frac12 x+13 & \text{if }x < 4 \end{cases}$$ I am trying to determine if the equations are differentiable at $x=...
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0 votes
0 answers
39 views

What do you call it when, in the limit, two functions are proportional by a factor of 1?

Two functions $f(x)$ and $g(x)$ have the property that $$\lim_{x\to\infty}\frac{f(x)}{g(x)}=1$$. What is this referred to as commonly? All I know of is Big-$\Theta$ notation, which says $$f(x)\in\...
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