# 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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### 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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### 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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### 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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### 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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### $f,g$ 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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### 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 ...
1 vote
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### 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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### 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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### 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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### 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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### 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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### 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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### 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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