Inverse Functions Help!! (1 Viewer)

[ludac]

Any Help Will be Appreciated. Thanks.

 

[random-1006]

Any Help Will be Appreciated. Thanks.

1. a f(-x)= sin ( cos^-1 (-x) ) = sin ( pi- cos^-1 x ) = sin ( cos^ -1 (x) ) { sin ( 180 -x) = sin(x) as we are in 2nd quadrant } = f(x). Thus even function.

b. range cos ^ (-1) x : 0 <= y <= pi, now looking at the sin graph, it is above the x axis between x = 0 and x=pi. which means were are putting values between 0 and pi INTO the sin function, the sin function will only GIVE out values >=0

c. y= sin ( cos ^ -1 (x) )
y^2 = [ sin ( cos ^ -1 (x) ) ] ^2, let u = cos ^ (-1) x , cos(u) =x
so we have y^2 = [sinu]^2, by the substitution
y^2 = 1-(cosu)^2= 1- (x^2) [ cosu = x], take sqrt both sides, we get + or -, but as we know from part b that it is >=0, we take the +

d.

just a semicircle, radius 1 , centred at the origin,its the top half of the circle x^2 +y^2 =1 .

e. similar to d

 

[ludac]

Thanks random-1006. But i am struggling with part d) from q14

 

[random-1006]

Thanks random-1006. But i am struggling with part d) from q14

i think its a series of lines, more specifically the line y=x, repeated every pi units.

ie. the line y= x ( from x= -pi/2 to pi/2) NOT including the end points, and then repeat that every pi units. Hopefully that makes some geometric sense.

remember later parts follow on from previous parts, look back to the previous parts, ok from part a you should have got " odd" as the answer ( ie we can rotate 180 degrees about origin and get same graph) , now for part b we have that the graph looks like y=x between -pi/2 and pi/2, and from part c, we know its periodic, ie repeats, the graph we must sketch will satisifies all these

to describe the graph a little more clearly it will look like "/////////// " with the greatest point y= pi/2 and the lowest point y= -pi/2 , ( but continues indefinately of course)ie when you are at the top of the left most line, you will move along to right and go to the very bottom of that line to continue. but all the points of the lines at the very top and very bottom will have the open circles ( as its not define there). Hopefully i have explained it good enough for you to draw the correct picture