Domain of Inverse Graph

[generalkorn12]

Hi,

Could anyone explain why a graph of, say, y = cos^-1(3x), all corresponding x-values are 3 times smaller (1 is now 1/3)?

Thanks.

[brightsky]

Hi,

Could anyone explain why a graph of, say, y = cos^-1(3x), all corresponding x-values are 3 times smaller (1 is now 1/3)?

Thanks.

think about the transformation involved when turning y=cos^(-1)(x) into y = cos^(-1)(3x). you dilate by a factor 1/3 from the y-axis. picture this on the x-y plane. any dilation from the y-axis changes the x-coordinate. hence (x,y) --> (1/3 x, y). this is why the domain is 1/3 of the original.

[VCE_2012]

Hi,

Could anyone explain why a graph of, say, y = cos^-1(3x), all corresponding x-values are 3 times smaller (1 is now 1/3)?

Thanks.

Now consider y=1/3cos(x):
- Domain=Real numbers
-Range=[-1/3,1/3]===> the range is 3 times smaller compared to y=cos(x)

As you should know: the domain of an inverse function is the same as the range of the 'original function'===> dom{f^-1(x)}=ran{f(x)}.
Therefore, the domain of y = cos^-1(3x) is: [-1/3,1/3]
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If you don't know already to work out the domain of the inverse function, here it is:

for y = cos^-1(x)

the domain= [-1,1]

therefore for y = cos^-1(3x)

the domain= -1<3x<1 (<,> represent greater/less or equal to in this case)
                --> -1/3<x<1/3
--->[-1/3,1/3]