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Draw the graph of f(x) = sqrt(1-e^(-x^(2...

Draw the graph of `f(x) = sqrt(1-e^(-x^(2))`

Text Solution

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We have `y=f(x) sqrt(1-e^(-x^(2))`
`f(x)` is defined if `1-e^(-x^(2)) ge0`
or `-x^(2) le0`
`x^(2) ge0`
Hence, the domain is R.
`f^(')(x) = (x.e^(-x^(2)))/sqrt(1-e^(-x^(2)))`
So, f(x) is increasing for `x gt 0` and decreasing for `x lt 0`.
Also `f^(')(0)` does not exist, so `f(x)` is non-differentiable at `x=0`.
`underset(x to +-infty)"lim" sqrt(1-e^(-x^(2)))=1`
So y=1 is an asymptote.
Further `sqrt(1-e^(-x^(2))) ge0, AAx in R.`
Thus, the graph of `y=f(x)` is as shown in the following figure.
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