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Draw the graph of y=cos^(-1)(2^(x))....

Draw the graph of `y=cos^(-1)(2^(x))`.

Text Solution

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We have `y=f(x) =y cos^(-1)(2^(x))`
Now for `x in R, 2^(x) in (0,oo)`
But f(x) is defined if `2^(x) in (0, 1]`,
So `x in (-oo, 0]," which is the domain of y=f(x)`
Now for `0lt2^(x)le1`
`cos^(-1)1lecos^(-1)(2^(x))ltcos^(-1)0`
`:." "0lecos^(-1)(2^(x))ltpi//2`
Thus, when `xto-oo,cos^(-1)(2^(x))to(pi)//2"and "f(0)=0`
Also `f'(x)=(-2^(x)log_(e)2)/(sqrt(1-4^(x)))lt0,` hence f(x) is a decreasing function.
From the above discussion, the graph of the function is as shown in the following figure.
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