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Relation and FunctionsQuestion and Answers: Page 1

Question Number 207314 Answers: 1 Comments: 0

let f:R→R be a continuous function then show that (1) if f(x) = f(x^2 ) ∀ x ∈R then f is a constant function (2) if f(x) = f(2x+1) ∀x∈R then f is a constant function

$$\:\mathrm{let}\:\mathrm{f}:\mathbb{R}\rightarrow\mathbb{R}\:\mathrm{be}\:\mathrm{a}\:\mathrm{continuous}\:\mathrm{function}\:\mathrm{then} \\ $$$$\:\mathrm{show}\:\mathrm{that} \\ $$$$\left(\mathrm{1}\right)\:\mathrm{if}\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{f}\left(\mathrm{x}^{\mathrm{2}} \right)\:\forall\:\mathrm{x}\:\in\mathbb{R}\:\mathrm{then}\:\mathrm{f}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant} \\ $$$$\:\:\mathrm{function} \\ $$$$\:\left(\mathrm{2}\right)\:\mathrm{if}\:\:\mathrm{f}\left(\mathrm{x}\right)\:=\:\mathrm{f}\left(\mathrm{2x}+\mathrm{1}\right)\:\forall\mathrm{x}\in\mathbb{R}\:\mathrm{then}\:\mathrm{f}\:\:\mathrm{is}\:\mathrm{a}\: \\ $$$$\:\:\mathrm{constant}\:\mathrm{function}\: \\ $$

Question Number 207085 Answers: 2 Comments: 0

Let f be a function with the following properties: (i) f(1) =1 (ii) f(2n)=n.f(n) for any positive integer n. Find the value of f(2^(10) ) a)1 b) 2^(10 ) c) 2^(35) d) 2^(45)

$${Let}\:{f}\:{be}\:{a}\:{function}\:{with}\:{the}\:{following} \\ $$$${properties}:\:\left({i}\right)\:{f}\left(\mathrm{1}\right)\:=\mathrm{1}\:\left({ii}\right)\:{f}\left(\mathrm{2}{n}\right)={n}.{f}\left({n}\right)\:{for} \\ $$$${any}\:{positive}\:{integer}\:{n}.\:{Find}\:{the}\:{value} \\ $$$${of}\:{f}\left(\mathrm{2}^{\mathrm{10}} \right) \\ $$$$\left.{a}\left.\right)\left.\mathrm{1}\left.\:{b}\right)\:\mathrm{2}^{\mathrm{10}\:} {c}\right)\:\mathrm{2}^{\mathrm{35}} \:{d}\right)\:\mathrm{2}^{\mathrm{45}} \\ $$

Question Number 206993 Answers: 1 Comments: 0

If the nth term of a sequence is given by ((n^2 −2n)/4) ,what is the sum of n terms of the sequence?

$${If}\:{the}\:{nth}\:{term}\:{of}\:{a}\:{sequence}\:{is} \\ $$$${given}\:{by}\:\frac{{n}^{\mathrm{2}} −\mathrm{2}{n}}{\mathrm{4}}\:,{what}\:{is}\:{the}\:{sum}\:{of}\:{n} \\ $$$${terms}\:{of}\:{the}\:{sequence}? \\ $$

Question Number 206442 Answers: 1 Comments: 0

Question Number 206151 Answers: 1 Comments: 0

Question Number 206136 Answers: 1 Comments: 0

Question Number 205919 Answers: 1 Comments: 0

write the following recursive function in explicit form f(1)=1 f(n+1)=(n+1)f(n)+n!

$$\mathrm{write}\:\mathrm{the}\:\mathrm{following}\:\mathrm{recursive}\:\mathrm{function}\:\mathrm{in}\:\mathrm{explicit}\:\mathrm{form} \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{1} \\ $$$${f}\left({n}+\mathrm{1}\right)=\left({n}+\mathrm{1}\right){f}\left({n}\right)+{n}! \\ $$

Question Number 205262 Answers: 1 Comments: 0

nature of the serie Σ_(n≥1) ((ln(n))/n)

$${nature}\:{of}\:{the}\:{serie}\:\sum_{{n}\geqslant\mathrm{1}} \:\frac{{ln}\left({n}\right)}{{n}} \\ $$

Question Number 204478 Answers: 1 Comments: 0

soit f: R^3 →R^3 f(x,y,z)=(x+y,2x−y,x+z) •1 Ecrire la matrice M de cette application dans la base canonique B de R^3 •2 Calculer f(1,2,3)de 2 manieres differentes −en utilisant la definition de f −en utilisant la matrice M •3 determiner bsse de Ker( f) et de Im(f) •4 soient v_1 =(1,1,0) v_2 =(1,2,1) v_3 =(1,3,1) Montrer que la famille E=(v_1 , v_2 , v_3 )est une base de R^3 •5Calculer f(v_1 ) donner ses coordonnes(locus) dans bass E avec f(v_2 )=v_1 +6v_2 −4v_3 f(v_3 )=2v_1 +8v_2 −6v_3 •6 Ecrire la matrice N de f dans base F •7 Retrouver cette matrice a partir de M en utilisant la formule de changement de base

$$\mathrm{soit}\:\boldsymbol{\mathrm{f}}:\:\:\mathbb{R}^{\mathrm{3}} \rightarrow\mathbb{R}^{\mathrm{3}} \:\:\:\boldsymbol{\mathrm{f}}\left(\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}},\boldsymbol{\mathrm{z}}\right)=\left(\mathrm{x}+\mathrm{y},\mathrm{2x}−\mathrm{y},\mathrm{x}+\mathrm{z}\right) \\ $$$$\bullet\mathrm{1}\:\:\mathrm{Ecrire}\:\mathrm{la}\:\mathrm{matrice}\:\mathrm{M}\:\mathrm{de}\:\mathrm{cette}\:\mathrm{application} \\ $$$$\:\:\:\mathrm{dans}\:\mathrm{la}\:\mathrm{base}\:\mathrm{canonique}\:{B}\:\mathrm{de}\:\:\mathbb{R}^{\mathrm{3}} \: \\ $$$$\bullet\mathrm{2}\:\:\mathrm{Calculer}\:\boldsymbol{\mathrm{f}}\left(\mathrm{1},\mathrm{2},\mathrm{3}\right)\mathrm{de}\:\mathrm{2}\:\mathrm{manieres}\:\mathrm{differentes} \\ $$$$\:−\mathrm{en}\:\mathrm{utilisant}\:\mathrm{la}\:\mathrm{definition}\:\mathrm{de}\:\mathrm{f} \\ $$$$−\mathrm{en}\:\mathrm{utilisant}\:\mathrm{la}\:\mathrm{matrice}\:{M}\: \\ $$$$\bullet\mathrm{3}\:\:\mathrm{determiner}\:\mathrm{bsse}\:\mathrm{de}\:\mathrm{Ker}\left(\:\boldsymbol{\mathrm{f}}\right)\:\mathrm{et}\:\mathrm{de}\:{I}\mathrm{m}\left(\boldsymbol{\mathrm{f}}\right) \\ $$$$\bullet\mathrm{4}\:\:\mathrm{soient}\:\mathrm{v}_{\mathrm{1}} =\left(\mathrm{1},\mathrm{1},\mathrm{0}\right)\:\mathrm{v}_{\mathrm{2}} =\left(\mathrm{1},\mathrm{2},\mathrm{1}\right)\:\:\mathrm{v}_{\mathrm{3}} =\left(\mathrm{1},\mathrm{3},\mathrm{1}\right) \\ $$$$\mathrm{Montrer}\:\mathrm{que}\:\mathrm{la}\:\mathrm{famille}\:{E}=\left(\mathrm{v}_{\mathrm{1}} ,\:\mathrm{v}_{\mathrm{2}} ,\:\mathrm{v}_{\mathrm{3}} \right)\mathrm{est} \\ $$$$\mathrm{une}\:\mathrm{base}\:\mathrm{de}\:\mathbb{R}^{\mathrm{3}} \\ $$$$\bullet\mathrm{5Calculer}\:\mathrm{f}\left(\mathrm{v}_{\mathrm{1}} \right)\:\mathrm{donner}\:\mathrm{ses}\:\mathrm{coordonnes}\left(\boldsymbol{\mathrm{locus}}\right) \\ $$$$\:\mathrm{dans}\:\mathrm{bass}\:{E} \\ $$$$\:\:\mathrm{avec}\:\mathrm{f}\left(\mathrm{v}_{\mathrm{2}} \right)=\mathrm{v}_{\mathrm{1}} +\mathrm{6v}_{\mathrm{2}} −\mathrm{4v}_{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{f}\left(\mathrm{v}_{\mathrm{3}} \right)=\mathrm{2v}_{\mathrm{1}} +\mathrm{8v}_{\mathrm{2}} −\mathrm{6v}_{\mathrm{3}} \\ $$$$\bullet\mathrm{6}\:\:\mathrm{Ecrire}\:\mathrm{la}\:\mathrm{matrice}\:{N}\:\mathrm{de}\:\boldsymbol{\mathrm{f}}\:\:\mathrm{dans}\:\mathrm{base}\:{F} \\ $$$$\bullet\mathrm{7}\:\:\mathrm{Retrouver}\:\mathrm{cette}\:\mathrm{matrice}\:\mathrm{a}\:\mathrm{partir}\:\mathrm{de}\:{M} \\ $$$$\mathrm{en}\:\mathrm{utilisant}\:\mathrm{la}\:\mathrm{formule}\:\mathrm{de}\:\mathrm{changement}\:\mathrm{de}\:\mathrm{base} \\ $$$$ \\ $$

Question Number 204426 Answers: 0 Comments: 0

let a , b >0 find all differentiable function f:(0,∞)→(0,∞) such that f′((a/x)) = ((bx)/(f(x))) , ∀ x>0

$$\:\:\mathrm{let}\:\mathrm{a}\:,\:\mathrm{b}\:>\mathrm{0}\:\:\mathrm{find}\:\mathrm{all}\:\mathrm{differentiable}\:\mathrm{function} \\ $$$$\:\:\:\mathrm{f}:\left(\mathrm{0},\infty\right)\rightarrow\left(\mathrm{0},\infty\right)\:\:\mathrm{such}\:\mathrm{that}\: \\ $$$$\:\:\:\:\mathrm{f}'\left(\frac{{a}}{\mathrm{x}}\right)\:\:=\:\:\frac{\mathrm{bx}}{\mathrm{f}\left(\mathrm{x}\right)}\:\:\:\:,\:\:\:\forall\:\mathrm{x}>\mathrm{0} \\ $$

Question Number 204394 Answers: 0 Comments: 0

Question Number 204393 Answers: 0 Comments: 0

Question Number 203905 Answers: 0 Comments: 8

let ABC a given triangle. Can we find three positions I,J,K on the side AB,AC,BC Such that IJK is equilateral?

$${let}\:{ABC}\:{a}\:{given}\:{triangle}.\:{Can}\:{we}\:{find} \\ $$$${three}\:{positions}\:{I},{J},{K}\:{on}\:{the}\:{side}\:{AB},{AC},{BC} \\ $$$${Such}\:{that}\:{IJK}\:{is}\:{equilateral}? \\ $$

Question Number 203291 Answers: 1 Comments: 0

f(x)={1+((√x^2 )/x) if x#0 2 if x=0 study the continuty of f in 0

$${f}\left({x}\right)=\left\{\mathrm{1}+\frac{\sqrt{{x}^{\mathrm{2}} }}{{x}}\:\:\:{if}\:{x}#\mathrm{0}\right. \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{2}\:\:{if}\:\:{x}=\mathrm{0} \\ $$$${study}\:{the}\:{continuty}\:{of}\:{f}\:{in}\:\mathrm{0} \\ $$

Question Number 202865 Answers: 3 Comments: 1

find the seauence u_n wich verify u_0 =1 and u_n +u_(n+1) =(((−1)^n )/n) for n≥1

$${find}\:{the}\:{seauence}\:{u}_{{n}} {wich}\:{verify} \\ $$$${u}_{\mathrm{0}} =\mathrm{1}\:{and}\:{u}_{{n}} +{u}_{{n}+\mathrm{1}} =\frac{\left(−\mathrm{1}\right)^{{n}} }{{n}} \\ $$$${for}\:{n}\geqslant\mathrm{1} \\ $$

Question Number 202795 Answers: 3 Comments: 0

if f(−1)=f(0)=f(2)=0 and f(1)=6 then find f(x)=?

$${if}\:{f}\left(−\mathrm{1}\right)={f}\left(\mathrm{0}\right)={f}\left(\mathrm{2}\right)=\mathrm{0}\:{and}\:{f}\left(\mathrm{1}\right)=\mathrm{6} \\ $$$${then}\:{find}\:{f}\left({x}\right)=? \\ $$

Question Number 202536 Answers: 2 Comments: 0

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Question Number 201848 Answers: 1 Comments: 0

Question Number 201681 Answers: 1 Comments: 0

f(x+1)−f(x)=3f(x)×f(x+1) D_f =N 2023×f(1402)=1 have equation f(x)=1 solution?

$${f}\left({x}+\mathrm{1}\right)−{f}\left({x}\right)=\mathrm{3}{f}\left({x}\right)×{f}\left({x}+\mathrm{1}\right) \\ $$$${D}_{{f}} ={N} \\ $$$$\mathrm{2023}×{f}\left(\mathrm{1402}\right)=\mathrm{1} \\ $$$${have}\:{equation}\:{f}\left({x}\right)=\mathrm{1}\:{solution}? \\ $$

Question Number 200973 Answers: 0 Comments: 0

Question Number 200722 Answers: 1 Comments: 1

Find all polynomials P(x) with real coefficients such that for all nonzero real numbers x, P(x)+P((1/x))=((P(x+(1/x))+P(x−(1/x)))/2)

$$\:\:\mathrm{Find}\:\mathrm{all}\:\mathrm{polynomials}\:\mathrm{P}\left(\mathrm{x}\right)\:\mathrm{with} \\ $$$$\:\mathrm{real}\:\mathrm{coefficients}\:\mathrm{such}\:\mathrm{that}\:\mathrm{for} \\ $$$$\:\mathrm{all}\:\mathrm{nonzero}\:\mathrm{real}\:\mathrm{numbers}\:\mathrm{x},\: \\ $$$$\:\:\:\:\:\mathrm{P}\left(\mathrm{x}\right)+\mathrm{P}\left(\frac{\mathrm{1}}{\mathrm{x}}\right)=\frac{\mathrm{P}\left(\mathrm{x}+\frac{\mathrm{1}}{\mathrm{x}}\right)+\mathrm{P}\left(\mathrm{x}−\frac{\mathrm{1}}{\mathrm{x}}\right)}{\mathrm{2}}\:\:\: \\ $$

Question Number 199781 Answers: 1 Comments: 0

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Question Number 199167 Answers: 0 Comments: 0

Give a function f: R→(0;+∞) continous on R and such that f(x+y) = f(x).f(y) a. Prove f(0) = 1 b. Let h(x) = ln[f(x)]. Prove that: h(x+y) = h(x) + h(y) c. Find all the function f such that problem request

$${Give}\:{a}\:{function}\: \\ $$$${f}:\:{R}\rightarrow\left(\mathrm{0};+\infty\right)\:{continous}\:{on}\:{R}\:{and}\:{such}\:{that} \\ $$$${f}\left({x}+{y}\right)\:=\:{f}\left({x}\right).{f}\left({y}\right) \\ $$$${a}.\:{Prove}\:{f}\left(\mathrm{0}\right)\:=\:\mathrm{1} \\ $$$${b}.\:{Let}\:{h}\left({x}\right)\:=\:{ln}\left[{f}\left({x}\right)\right].\:{Prove}\:{that}: \\ $$$$\:{h}\left({x}+{y}\right)\:=\:{h}\left({x}\right)\:+\:{h}\left({y}\right) \\ $$$${c}.\:{Find}\:{all}\:{the}\:{function}\:{f}\:{such}\:{that}\:{problem}\:{request} \\ $$$$\:\:\: \\ $$$$\: \\ $$

Question Number 198178 Answers: 2 Comments: 0

f(xf(y)+x)=xy+f(x) f:R→R f(x)=?

$${f}\left({xf}\left({y}\right)+{x}\right)={xy}+{f}\left({x}\right) \\ $$$${f}:\mathbb{R}\rightarrow\mathbb{R} \\ $$$${f}\left({x}\right)=? \\ $$

Question Number 198065 Answers: 1 Comments: 0

2^x +9+2^x =40

$$\mathrm{2}^{{x}} +\mathrm{9}+\mathrm{2}^{{x}} =\mathrm{40} \\ $$

Question Number 198064 Answers: 1 Comments: 0

3×5^x +5^(x+1) =8×5^3

$$\mathrm{3}×\mathrm{5}^{{x}} +\mathrm{5}^{{x}+\mathrm{1}} =\mathrm{8}×\mathrm{5}^{\mathrm{3}} \\ $$

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