inverse trig (1 Viewer)

[VJ30]

can someone show me a method of findin domain and range and sketching graph of these functions

y= arccos(cosx)

y=cos(arccosx)

y=arctan(tanx).....thanks in advance

 

[Carrotsticks]

can someone show me a method of findin domain and range and sketching graph of these functions

y= arccos(cosx)

y=cos(arccosx)

y=arctan(tanx).....thanks in advance

y= arccos(cosx)

D: All real x

R: y E [-1,1]

y=cos(arccosx)

D: x E [-1,1]

R: y E [-1,1]

y=arctan(tanx)

D: All real x

R: y E (- pi/2, pi/2)

-------------------------------------------------

Notation:

x E [a,b] means a <= x <= b

and

x E (a,b) means a < x < b

 

[Carrotsticks]

Oh I just saw you wanted an explanation, somebody may have to do this for me, a bit busy now.

If nobody does by tomorrow, I'll finish it off.

 

[VJ30]

Oh I just saw you wanted an explanation, somebody may have to do this for me, a bit busy now.

If nobody does by tomorrow, I'll finish it off.

thanks anyways u did clear some of my doubts but how did u get all real x as domain for the first one

 

[deswa1]

thanks anyways u did clear some of my doubts but how did u get all real x as domain for the first one

Think about it like this. Is there any value of x that you can sub in that makes the expression undefined? cosx is defined for every value of x and will give a value between -1 and 1. arccos is defined between -1 and 1, therefore all values of x will give a valid value (this isn't phrased well) that will allow arccos(cosx) to be defined.

 

[Sy123]

On Sketching, you can simply differentiate, and analyse the gradient of the curve at different values of x.
If that isnt enough, you can graph the differentiated function, then integrate "graphically"
Eg.

So in this case, whenever sine is negative, that is a negative gradient according to the f'(x).
Once you gather the information, that the gradient = 1 between certain intervals (1st quadrant, 2nd quadrant, i.e. Whenever sinx>0) and when the gradient =-1. You are then able to generally sketch the curve shape.
To get the range of the curve, look at the outermost function and see what the limitations there are.
For example for a)

The outermost function is acos, which ranges from 0 to pi. This is true no matter what inner function it is. The inner function of course changes the trajectory, but will never change the range of the curve.

Using these two pieces of information, you can sketch any curve really.

 

[Drongoski]

For 1st one: cosx can take any real value of x (its value would go from -1 to +1 to -1 ... repeatedly) and range will be [0,pi] if I'm still awake. And if I'm still awake, the range for the last one is (-pi/2, pi/2) rather than [-pi/2. pi/2]

 

[VJ30]

thanks heaps everyone!!!

 

[Carrotsticks]

For 1st one: cosx can take any real value of x (its value would go from -1 to +1 to -1 ... repeatedly) and range will be [0,pi] if I'm still awake. And if I'm still awake, the range for the last one is (-pi/2, pi/2) rather than [-pi/2. pi/2]

Yep, didn't change brackets when I copy/pasted.