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Relation and FunctionsQuestion and Answers: Page 1

Question Number 221413 Answers: 7 Comments: 0

Question Number 221154 Answers: 0 Comments: 1

f(x)= (x/(∣ x ∣ + 1)) f(f(f(f(x)))) =?

$$\:{f}\left({x}\right)=\:\frac{{x}}{\mid\:{x}\:\mid\:+\:\mathrm{1}} \\ $$$$\:\:{f}\left({f}\left({f}\left({f}\left({x}\right)\right)\right)\right)\:=? \\ $$

Question Number 221153 Answers: 1 Comments: 1

f(x)= (1/2^x ) + (1/3^x ) + (1/4^x ) + ... +(1/(4000^x )) f(2) + f(3) + f(4)+ ... =?

$$\:\:{f}\left({x}\right)=\:\frac{\mathrm{1}}{\mathrm{2}^{{x}} }\:+\:\frac{\mathrm{1}}{\mathrm{3}^{{x}} }\:+\:\frac{\mathrm{1}}{\mathrm{4}^{{x}} }\:+\:...\:+\frac{\mathrm{1}}{\mathrm{4000}^{{x}} } \\ $$$$\:\:{f}\left(\mathrm{2}\right)\:+\:{f}\left(\mathrm{3}\right)\:+\:{f}\left(\mathrm{4}\right)+\:...\:=? \\ $$

Question Number 220869 Answers: 1 Comments: 1

Find the maximum value of x^2 y^3 z^4 subject to the condition x+y+z=18

$${Find}\:{the}\:{maximum}\:{value}\:{of}\:{x}^{\mathrm{2}} {y}^{\mathrm{3}} {z}^{\mathrm{4}} \:{subject}\:{to}\:{the}\:{condition}\:{x}+{y}+{z}=\mathrm{18} \\ $$

Question Number 219733 Answers: 2 Comments: 0

Question Number 219377 Answers: 1 Comments: 0

If ((fog)^(−1) of)(x)= 3x−8 find g(5).

$$\:\mathrm{If}\:\left(\left(\mathrm{fog}\right)^{−\mathrm{1}} \mathrm{of}\right)\left(\mathrm{x}\right)=\:\mathrm{3x}−\mathrm{8} \\ $$$$\:\mathrm{find}\:\mathrm{g}\left(\mathrm{5}\right). \\ $$

Question Number 219090 Answers: 2 Comments: 0

Prove that the sequence a_n =(1/( ((n!))^(1/n) )) is decreasing.

$$\mathrm{Prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{sequence}\:{a}_{{n}} =\frac{\mathrm{1}}{\:\sqrt[{{n}}]{{n}!}}\:\mathrm{is}\:\mathrm{decreasing}. \\ $$

Question Number 218673 Answers: 3 Comments: 1

Question Number 218311 Answers: 1 Comments: 0

Question Number 218256 Answers: 2 Comments: 0

Question Number 217725 Answers: 0 Comments: 2

Question Number 216077 Answers: 1 Comments: 0

Find the largest value of the non negative integer p for which lim_(x→1) {((− px + sin(x − 1) + p)/(x + sin(x − 1) − 1))}^((1 − x)/(1 − (√x))) = (1/4) .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{largest}\:\mathrm{value}\:\mathrm{of}\:\mathrm{the}\:\mathrm{non}\:\mathrm{negative} \\ $$$$\mathrm{integer}\:{p}\:\mathrm{for}\:\mathrm{which}\: \\ $$$$\underset{{x}\rightarrow\mathrm{1}} {\mathrm{lim}}\:\left\{\frac{−\:{px}\:+\:\mathrm{sin}\left({x}\:−\:\mathrm{1}\right)\:+\:{p}}{{x}\:+\:\mathrm{sin}\left({x}\:−\:\mathrm{1}\right)\:−\:\mathrm{1}}\right\}^{\frac{\mathrm{1}\:−\:{x}}{\mathrm{1}\:−\:\sqrt{{x}}}} \:=\:\frac{\mathrm{1}}{\mathrm{4}}\:. \\ $$

Question Number 215887 Answers: 0 Comments: 2

Question Number 215331 Answers: 1 Comments: 0

Question Number 214644 Answers: 1 Comments: 0

((−6)/7)/((−7)/6)

$$\frac{−\mathrm{6}}{\mathrm{7}}/\frac{−\mathrm{7}}{\mathrm{6}} \\ $$

Question Number 213291 Answers: 1 Comments: 0

Find domain of y_(213291) : y_(213291) =((3+e^((x^2 −3x+2)/(x−6)) )/(log_(3/4) (√(x^2 −(1/4)))))

$${Find}\:{domain}\:{of}\:{y}_{\mathrm{213291}} : \\ $$$${y}_{\mathrm{213291}} =\frac{\mathrm{3}+{e}^{\frac{{x}^{\mathrm{2}} −\mathrm{3}{x}+\mathrm{2}}{{x}−\mathrm{6}}} }{\mathrm{log}_{\frac{\mathrm{3}}{\mathrm{4}}} \sqrt{{x}^{\mathrm{2}} −\frac{\mathrm{1}}{\mathrm{4}}}} \\ $$

Question Number 212518 Answers: 2 Comments: 0

⇃

$$\:\:\:\cancel{\underbrace{\downharpoonleft}\underline{}\:} \\ $$

Question Number 209707 Answers: 0 Comments: 0

a,b ∈C : ab^− + b = 0 f : z′ = az^− + b such that f(M) = M′ 1. let z_A = z and z_(A′) = z′ and f(A) = A show that 2Re(b^− z) = bb^− (A is the set of invariant points and describes a line (△) ) 2. Deduce that (△) is a line with gradient u^( →) with affix z_u^→ = ib 3. show that (z_(MM ′) /z_u ) = ((bb^− − 2Re(bz^− ))/(ibb^− )) 4. show that 2Re(b^− z_0 ) = bb^_ where z_0 = ((z + z ′)/2) 5. Deduce that for M ∉ (△) , M is a perpendicular bisector of [MM ′]

$${a},{b}\:\in\mathbb{C}\::\:{a}\overset{−} {{b}}\:+\:{b}\:=\:\mathrm{0}\:{f}\::\:{z}'\:=\:{a}\overset{−} {{z}}\:+\:{b}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:{such}\:{that}\:{f}\left({M}\right)\:=\:{M}' \\ $$$$\mathrm{1}.\:{let}\:{z}_{{A}} \:=\:{z}\:{and}\:{z}_{{A}'} \:=\:{z}'\:{and}\:{f}\left({A}\right)\:=\:{A} \\ $$$${show}\:{that}\:\mathrm{2}{Re}\left(\overset{−} {{b}z}\right)\:=\:{b}\overset{−} {{b}} \\ $$$$\left({A}\:{is}\:{the}\:{set}\:{of}\:{invariant}\:{points}\:{and}\right. \\ $$$$\left.\:{describes}\:{a}\:{line}\:\left(\bigtriangleup\right)\:\right) \\ $$$$\mathrm{2}.\:{Deduce}\:{that}\:\left(\bigtriangleup\right)\:{is}\:{a}\:{line}\:{with}\: \\ $$$${gradient}\:\overset{\:\rightarrow} {{u}}\:{with}\:{affix}\:{z}_{\overset{\rightarrow} {{u}}} \:=\:{ib} \\ $$$$\mathrm{3}.\:{show}\:{that}\:\frac{{z}_{{MM}\:'} }{{z}_{{u}} }\:=\:\frac{{b}\overset{−} {{b}}\:−\:\mathrm{2}{Re}\left({b}\overset{−} {{z}}\right)}{{ib}\overset{−} {{b}}} \\ $$$$\mathrm{4}.\:{show}\:{that}\:\mathrm{2}{Re}\left(\overset{−} {{b}z}_{\mathrm{0}} \right)\:=\:{b}\overset{\_} {{b}}\:{where} \\ $$$$\:{z}_{\mathrm{0}} \:=\:\frac{{z}\:+\:{z}\:'}{\mathrm{2}} \\ $$$$\mathrm{5}.\:{Deduce}\:{that}\:{for}\:{M}\:\notin\:\left(\bigtriangleup\right)\:,\:{M}\:{is}\: \\ $$$${a}\:{perpendicular}\:{bisector}\:{of}\:\left[{MM}\:'\right] \\ $$

Question Number 208533 Answers: 2 Comments: 0

z′ = (1/2)(z+(1/z)) z and z′ are complex numbers show that z = 2e^(iθ) show that M′ describes a conic section

$${z}'\:=\:\frac{\mathrm{1}}{\mathrm{2}}\left({z}+\frac{\mathrm{1}}{{z}}\right) \\ $$$${z}\:{and}\:{z}'\:{are}\:{complex}\:{numbers} \\ $$$${show}\:{that}\:{z}\:=\:\mathrm{2}{e}^{{i}\theta} \\ $$$${show}\:{that}\:{M}'\:{describes}\:{a}\:{conic}\:{section} \\ $$

Question Number 208431 Answers: 2 Comments: 0

h_a (x) = e^(−x) + ax^2 show that h_a admits a minimum in R

$${h}_{{a}} \left({x}\right)\:=\:{e}^{−{x}} \:+\:{ax}^{\mathrm{2}} \\ $$$${show}\:{that}\:{h}_{{a}} \:{admits}\:{a}\:{minimum}\:{in}\:\mathbb{R} \\ $$

Question Number 208418 Answers: 1 Comments: 0

u_(n+1) = u_n −u_n ^3 ; u_0 ∈ ]0, 1[ . show that u_n ∈ ]0, 1[ . show that u_n converges to 0 v_n = (1/u_(n+1) ^2 ) − (1/u_n ^2 ) . express v_n interms of u_n . show that v_n converges to 2 f(x) = ((2−x)/((1−x)^2 )) . show that f is increasing and deduce that v_n is decreasing . show that v_n ≥ 2

$$\left.{u}_{{n}+\mathrm{1}} \:=\:{u}_{{n}} −{u}_{{n}} ^{\mathrm{3}} \:;\:{u}_{\mathrm{0}} \:\in\:\right]\mathrm{0},\:\mathrm{1}\left[\right. \\ $$$$\left..\:{show}\:{that}\:{u}_{{n}} \:\in\:\right]\mathrm{0},\:\mathrm{1}\left[\right. \\ $$$$.\:{show}\:{that}\:{u}_{{n}} \:{converges}\:{to}\:\mathrm{0} \\ $$$${v}_{{n}} \:=\:\frac{\mathrm{1}}{{u}_{{n}+\mathrm{1}} ^{\mathrm{2}} }\:−\:\frac{\mathrm{1}}{{u}_{{n}} ^{\mathrm{2}} } \\ $$$$.\:{express}\:{v}_{{n}} \:{interms}\:{of}\:{u}_{{n}} \\ $$$$.\:{show}\:{that}\:{v}_{{n}} \:{converges}\:{to}\:\mathrm{2} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{2}−{x}}{\left(\mathrm{1}−{x}\right)^{\mathrm{2}} } \\ $$$$.\:{show}\:{that}\:{f}\:{is}\:{increasing}\:{and}\:{deduce}\:{that}\: \\ $$$${v}_{{n}} \:{is}\:{decreasing} \\ $$$$.\:{show}\:{that}\:{v}_{{n}} \:\geqslant\:\mathrm{2} \\ $$

Question Number 208103 Answers: 1 Comments: 0

Find inf{(m/n) ∣ m, n ∈ N, m<n−2}

$$\mathrm{Find}\:\mathrm{inf}\left\{\frac{{m}}{{n}}\:\mid\:{m},\:{n}\:\in\:\mathbb{N},\:{m}<{n}−\mathrm{2}\right\} \\ $$

Question Number 207390 Answers: 0 Comments: 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?

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

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