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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13 views

How to find a function that gives me the indexes of a square matrix

Let a square matrix M with size m, and let i and j, given $0 \leq i = j < m$, so that $a_{ij}$ is a value of the matrix located in the row i and column j (assuming that we start counting with 0 ...
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1answer
14 views

How to customize a function with a horizontal asymptote?

I'm looking for a function that approaches a y-value as x approaches infinity, something close to: $$y=\frac{x}{x+1}$$ however I have no idea how to customize such a function to suit my needs. I know ...
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1answer
26 views

If $\: f:A\to B\:\text{ with }A⊇\{x\},\{y\}\quad$then is $\:f(x)\equiv f(y\mapsto x)\:\text{ & }\:f(y)\equiv f(x\mapsto y)$?

If two variables $x$ and $y$ are both known to be in the domain of a unary function $f$ (but not necessarily defined over the whole domain nor be equal to the other; i.e., $A∋x,y$ while $◇\:\{x\}≠\{y\}...
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2answers
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When we are computing limit, why the point a(when x approaches a) we plugged in don’t necessarily need to be in the domain of definition

Like this equation : $$\lim _{x\to 0 }x\sin \frac{1}{x}$$ $x$ is the denominator, but we can nevertheless plug $x=0$ into the function to get the limit, why?
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1answer
37 views

Even function of a graph with local maximum at x-axis but no local minima

I am having some problems with this question. Question Give a possible equation of a function that: has a single local maximum, located at the x-axis, has no local minima, is an even function, and ...
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$f(yf^2(x))=x^3f(xy)$ for all $x,y \in {\mathbb{Q}}^{+}$

Find all functions $f: {\mathbb{Q}}^{+} \rightarrow {\mathbb{Q}}^{+}$ such that: $f(yf^2(x))=x^3f(xy)$ for all $x,y \in {\mathbb{Q}}^{+}$ My progress: $P(x;1) \Rightarrow f(f^2(x))=x^3f(x)$ Easily ...
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If a function $f$ is not defined on $[-2,0]$, what can we say about $\lim_{x\to-2}f(x)$?

I was thinking of a strange case in which we can say the limit is undefined. ( I mean different than the limit does not exist.) So I generated the graph of a piecewise function $f(x)$. There is a ...
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Comparison of $x^aa^x$ and $ x^x$

I need to compare values of $x^2a^x$ and $x^x$ at infinity. approach : I took f(x) = $(\frac{a}{x})^xx^a$ I found f’(x). I got information that f is increasing from 0 to a then decreasing after wards. ...
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What does the graph 3 represents?pls give answer in very detailed manner

enter image description here This a question from maths contest
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Recursive Definitions and Proposition 2.1.16 in Tao's Analysis I

In his book, Analysis I, Dr. Tao introduces Proposition 2.1.16: I think I know how to prove that each $a_n$ only takes one value and it is never "redefined," by induction: As the base case,...
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Is it okay if a function has two different points that have the exact same coordinates?

In math class we are learning about functions. One of the "stretch" questions is as follows: "Find the conditions for $a$ and $b$ that make $\{(a,b)(-a,b)(2a,b)(a^2,b)\}$ a function&...
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1answer
35 views

Why do we have "... for all $x\in\mathbb{R}$" when we define a function after also giving a domain?

I was reading about functions on this wikipedia page and in the section titled 'Notation', they give a proper example definition of a function: Let $f:\mathbb{R}\to\mathbb{R}$ be the function defined ...
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Functional equation $\left(f(x)-f(y)\right)f\left(\frac{x+y}2\right)-f(\sqrt{xy}))=0$ [closed]

Select all the continuous functions $f:[0,\infty)\to\mathbb{R}$ that satisfies the following equation $$\left(f(x)-f(y)\right)f\left(\frac{x+y}2\right)-f\left(\sqrt{xy}\right)=0$$ for each non-...
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Would it be a closed or open interval about infinity in this function [closed]

If [cot−1 x]+[cos-1 x]=0, range of the x would be (cos1,1] U (cot1, infinity )or] The interval including infinity would be a closed one or open one and why Please a little help here
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1answer
12 views

Has the associative property been generalized to k-ary functions?

I've been exploring why the associative property is so interesting to mathematicians. Along the way, I have found the rather obvious fact that it only works on binary operations. It needs a concept ...
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58 views

If $f(x)=1+4x-x^2$ $\;\forall \; \;x \in R$

If $f(x)=1+4x-x^2$ $\;\forall \; \;x \in R$ $g(x)=\left\{ \begin{array}{ll} \max f(t); \;x\leq t\leq x+1; \;0\leq x \lt3 \\ {6}; \; \mbox{if } 3\leq x\leq 5 \end{array} \right.$ Verify ...
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1answer
71 views

What may be true about the roots of the equation $ax^3 + bx -1 = 0$? (a and b are rational numbers)

Options are as follows It has 2 irrational and 1 rational root It has 1 irrational and 2 rational roots A ll its roots are complex None of the above As per the answer should be D. Let $a = 1, b = 0$ ...
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1answer
97 views

Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $f(x)=f(x^2+1).$

Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $f(x)=f(x^2+1).$ My tried: If $x \le 1,$ we consider the sequence $x_0=a\le 1$ $x_{n+1}=x_n^2 +1$ $\...
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89 views

Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $f(x-y)+f(y-z)+f(z-x)+27=0.$

Find all functions $f:\mathbb{R} \to \mathbb{R},$ which is continuous in $\mathbb{R}$ then $$f(x-y)+f(y-z)+f(z-x)+27=0.$$ I actually don't have any ideas to deal with it, but here is some tries: Let $...
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33 views

Submultiplicity when dealing with fractions

I understand that if $f: A \to B$ is a submultiplicative function, then $\exists K : f(ab)\leq K f(a)f(b)$ for every $(a,b)\in A\times A$. Now at some point, by arguing submultiplicity my teacher said ...
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1answer
22 views

Need help finding the equation of a cubic polynomial

I need help finding the equation of a real cubic polynomial that cuts the x-axis at 1/2 and -3, has a y-intercept of 30, and passes through (1, -20). So far I have a(2x-1)(x+3)(bx-c)? Is there a way ...
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41 views

Approximating an exponential function with another function

I have a function $f(\ell)=\exp\Big(-{\frac{a}{\ell b+c}}\Big)$. I would like to rewrite this function in the form $(1-d)g^{\ell}$. How can I express $g$ and $d$ in terms of $a$, $b$ and $c$? Here, a,...
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$\frac{\textrm{d}\quad\:}{\textrm{d}\big(x\:\:\text{ s.t. }{x=0}\big)}\Big(|x|\Big):=\text{[undefined]}$. But if $x:∈ℝ\:$→¿◇∃ a ‘best-fit’ derivative?

Let $y=:f(x)\:=|x|\quad\text{s.t.}\:f:ℝ→ℝ$. Under this defined reals-to-reals mapping (graphable on a $ℝ_x×ℝ_y$ cartesian grid, visibly symmetric with window constraints for $y≥0$ centered about $x=0$...
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3answers
105 views

When is the product of two injective functions, also injective?

I'm currently, by curiosity, investigating when the product of two injective functions, is also injective. My condition is that this is true if and only if there $\nexists x_1, x_2 \in X$ such that $f(...
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1answer
54 views

extension Rolle's theorem for limit values

Rolle's theorem states that: If a real-valued function $f$ is continuous on a proper closed interval $[a, b]$, differentiable on the open interval $(a, b)$, and $f (a) = f (b)$, then there exists at ...
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1answer
62 views

Find the minimum of $a^4+b^4+c^4+d^4+a^2+b^2+c^2+d^2$.

Let $a,b,c,d$ be real numbers such that $a+b+c+d=0$ and $abcd=1$. Find the minimum value of $a^4+b^4+c^4+d^4+a^2+b^2+c^2+d^2$. By $\text{Vieta}$'s theorem, $a,b,c,d$ are the roots of the equation $x^...
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28 views

Understanding a question from the topic FUNCTIONS.

I was trying to solve some problems about functions when I got stuck with this problem. I am unable to understand this. First and foremost, $$A=[1,2]$$ $$B=[3,4]$$ This seems to be a little daunting. ...
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1answer
65 views

Finding f(g) from f(x) and g(x)

I'm studying an article where there are two functions: $e(x) = (e_0/8)\cdot[(2x^3 + x)\sqrt{(1 + x^2)} − \sinh^{−1}(x)]$ $p(x) = (e_0/24)\cdot[(2x^3 - 3x)\sqrt{(1 + x^2)} + 3\sinh^{−1}(x)]$ The ...
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22 views

Functional equation $x f(y) -y f(x) =f(\dfrac{y}{x})$ [duplicate]

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R},x\neq 0$ then $$x f(y) -y f(x) =f(\dfrac{y}{x}).$$ My try: Let $x=1$ then $f(y) -y f(1) = f(y) \Rightarrow f(1) =...
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2answers
42 views

Solve a quartic function with three unknowns and two given roots.

I'd like to know how I could solve the following quartic function: $p(z)=2z^4+az^3+bz^2+cz+3$ given that it $2$ and $i$ should be part of their roots. I thought I should maybe be trying to turn this ...
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1answer
41 views

Formal Definition of the Product of Two Sets

I am taking a course in Algebraic Structures, and the notion of product of sets (a.k.a. Caratesian Product) came in. We were given a definition that made me feel I didn't understand it. They gave the ...
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2answers
38 views

$F(x, y) = \frac{x^2+y^2} {|x|+|y|} $ at $(x, y) \neq (0, 0)$ find if function is continuous or not? [closed]

$F(x, y) = \frac{x^2+y^2}{|x|+|y|}$ at $(x, y) \neq (0, 0)$ find if function is continuous or not? Continuity and Functions We have to tell whether function is continuous or not. I don't know how to ...
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0answers
29 views

What functions with source and target the rational numbers satisfy the intermediate value theorem?

I am curious to find if there is a characterization of the set $S$ of all functions $f: \mathbb{Q} \to \mathbb{Q}$ where for all intervals $[a,b] \subset \mathbb{Q}$, there exists $x \in [a,b]$ for ...
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I'm stuck with this proof: [closed]

Proof that -> $\lim\limits_{x\rightarrow 0^+}f(x) = \infty$ if, and only if $\lim\limits_{x\rightarrow \infty}f(\frac{1}{x}) = \infty$. Thanks everyone :D
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1answer
74 views

Find the number of injective functions $f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}$ such that $f(i)>f(i+1)$

Let $n$ and $m$ be two natural numbers, $m\ge n\ge 2$ . Find the number of injective functions $f\colon\{1,2,\ldots,n\}\to\{1,2,\ldots,m\}$ such that there exists a unique number $i\in\{1,2,\ldots,n-1\...
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2answers
35 views

Doubt proving that $(\forall f,f' \quad g\circ f = g\circ f' \implies f=f' ) \implies g \text{ is injective}$?

I am trying to understand the proof of: (ii) The function $g : b \to c$ is mono iff the following condition holds: for any two functions $f,f' : a \to b$, $g \circ f = g \circ f'$ implies $f=f'$ ...
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0answers
44 views

Hollow Square Bracket Notation

This function appeared on my HW but I do not recognize the bracket symbols. I am thinking they might be an equivalent notation for absolute value, but I'm not sure and would appreciate any ...
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2answers
89 views

Looking for function which satisfies $f(n)=f(2n)+f(2n+1)$

I am looking for a function from $\Bbb N^*$ to $\Bbb R^{+*}$ such that $f(n)=f(2n)+f(2n+1)$ (for any $n$ in $\Bbb N^*$). I also am looking for something smooth, where $f(n+1)-f(n)$ would be strictly ...
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1answer
44 views

Uniform convergence of a series of functions with arctan

I'm trying to solve the following problem about the uniform convergence of a series of functions involving the function arctan(x). $$ \sum_{i=1}^\infty \left[1-\frac{n}{x^{2n}}\arctan\left(\frac{x^{2n}...
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1answer
24 views

How can you determine where two polynomials cross and are tangent to each other, just by factorising the difference?

My textbook asked me to factorise the difference between P(x) and Q(x) and describe the intersection. I'm having trouble understanding: Why doing this would allow me to describe the intersection And ...
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2answers
30 views

The graph of a polynomial function of degree n is completely determined by any n + 1 points on the curve

Highschool maths student here, I'm having trouble understanding this statement: "The graph of a polynomial function of degree n is completely determined by any n + 1 points on the curve" Any ...
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0answers
41 views

Finding the value of $\lim_{x\to\infty}\frac{\int_0^x\cos^{-1}(\cos t)dt}{\int_0^x\sqrt{\{t\}-\{t\}^2}dt}$

Let $f:\mathbb R\to\mathbb R$ and $g:\mathbb R\to\mathbb R$ be defined as $f(x)=\cos^{-1}(\cos x)$ and $g(x)=\sqrt{\{x\}-\{x\}^2}$ where $\{x\}$ denotes fractional part of $x$. Find the value of $$\...
2
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1answer
46 views

Finding the value of $\displaystyle\int_{x_1+x_2}^{3x_2-x_1}\big\{\frac x4\big\}\left(1+\left[\tan\left(\frac{\{x\}}{1+\{x\}}\right)\right]\right)dx$

If $x_1$ and $x_2$ ($x_1\lt x_2$) are two values of $x$ satisfying the equation $\left|2\left(x^2+\frac1{x^2}\right)+|1-x^2|\right|=4\left(\frac32-2^{x^2-3}-\frac1{2^{x^2+1}}\right)$ then find the ...
3
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2answers
72 views

$f:[1,4]\rightarrow[7,14]$ is a concave surjective function then prove that $(f'(x))^2=49/9$ has at least one and at most two roots in $[1,4]$

$f:[1,4]\rightarrow[7,14]$ is a strictly concave surjective function then prove that $(f'(x))^2=49/9$ has at least one and at most two roots in $[1,4]$ $\displaystyle f'(x)=\pm\frac{7}{3}$, so if we ...
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0answers
32 views

Integration of unknown functions

Let $ \ f(x) \ $ and $ \ g(x) \ $ be continous positive function such that $ \ f(-x)=g(x)-1 \ $ and $ \ f(x)=g(x)/g(-x) \ . $ Integration of $ \ f(x) \ $ from -20 to 20 is 2020 . Find integration of $ ...
3
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2answers
112 views

If $y=f(x)$ is a concave upward function and $y=g(x)$ is a function such that $f'(x)g(x)-g'(x)f(x)=x^4+2x^2+10$ then prove that...

If $y=f(x)$ is a concave upward function and $y=g(x)$ is a function such that $f'(x)g(x)-g'(x)f(x)=x^4+2x^2+10$ then prove that $g(x)$ has at least one root between consecutive roots of $f(x)=0$ $f'(...
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1answer
28 views

Help me with this demonstration involving function composition [closed]

Given two functions $f, g : \mathbb{R}\to\mathbb{R}$, we define the composition of $f$ and $g$, denoted by $f\circ g$, as $$(f\circ g)(x) := f(g(x))$$ for all $x\in\mathbb{R}$. Suppose $g$ is ...
2
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1answer
38 views

how to obtain the strong convexity inequality

I was playing around with the strong convexity definition and got stuck at some point. I was wondering if someone could kindly help me out. We say that function $f$ is strongly convex if $1) f(x) \geq ...
2
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3answers
73 views

Using Lagrange multiplier , find minimum value of $xy(x^2 + y^2) +4$ , given that $x^2 + y^2 +xy -1 = 0 , x,y \in \mathbb R$

Using Lagrange multiplier , find minimum value of $$f(x,y)=xy(x^2 + y^2) +4$$ , Given that $g(x,y)=x^2 + y^2 +xy -1 = 0 , $for all values of $ x,y \in \mathbb R$. My attempt So i formed a function $$...

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