pls solve these sums for me in detail:
1) What is the period of the function
|sin2x|+|cos8x|
2) If f(x)=ln(x2+ex2+1),then find the range of f(x)
3) Let f:R→R be defined by f(x)=(ex-e-x)/2.Is f(x) invertible?.If so find its inverse.
4) The function f:R→R where R is a set of real numbers, is defined by f(x)=αx2+6x-8α+6x-8x2 .Find the interval of values of α for which f is onto.Is the function one to one for α=3?.Justify your answer.
(cudn't even understand wat the ques meant)
11) pi/2
3)it is invertible because it an bijective funn
let ex=t;
y=t2-1/2t;
so t2-2ty-1=0;
t=(2y±√4y2+4)/2;
replacin t
ex=(2y±√4y2+4)/2;
take ln on both side
inverted f(x) is
y=ln( (2x±√4x2+4)/2);
i cudnt understand the font of question 2 & 4
1For no 1)..My answer is always comin as π(pi).Cud u pls kindly solve it for me.here is a brief explanation of the method dat i'm following.
period of sinx=2π Thus peiod of sin2x i.e sin(2x+0)=2π2=π (if f(x)is a func wid period T then f(ax+b) is a a periodic func with period T/a)Similarly for cos8x priod is comin as π4. Hence L.C.M= π(Ans)
1for no(2) if f(x)=loge(x2+e/x2 +1).........in the numerator it is x2+e and in the denominator it is x2+1 the whole expression inside loge i.e ln.
For no (4): I am jus explaining the equation.Rest of the language i hope u can understand.
f(x)= α(alpha)x2+6x-8 .(This expression is in da numerator.) In the denominator→ α+6x-8x2 .[in each of the nmerator & denominator there is a (minus)sign b4 8].Finally it looks like dis:
f(x)=αx2+6x-8/α+6x-8x2 .Now pls solve da ques
3arpan bannerjee .....
dude period of |sinx| = pi ........... so sin2x = pi/2 .....
similarly for cosx .... its pi/8 .............
take LCM ...... . and then divide by two .....[becuz both are even functions]
32......
(e-y)/(y-1) = x^2 ....
(e-y)(y-1) >0
(y-e)(y-1)<0
therefore ..... y ε (1,e] ...
therefore .. ln y ε (0,1]
