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Consider f: R+-> [4," "oo) given by f(x)...

Consider `f: R_+-> [4," "oo)` given by `f(x)=x^2+4` . Show that f is invertible with the inverse `f^(-1)` of given f by `f^(-1)(y)=sqrt(y-4)` where `R_+` is the set of all non-negative real numbers.

Text Solution

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In `f: R_+ to [4, oo[, f(x) = x^(2) + 4`
Let `x, y in R_+ and " " f(x) =f(y)`
`rArr " "x ^(2) + 4 = y^(2) + 4`
`rArr " "x ^(2) = y^(2)`
`rArr " " x = y ( because xy in R_+)`
`therefore f `is one -one.
Again let `f(x) = y ` where `y in [ 4, oo[`
`rArr " " x^(2) + 4 =y`
`rArr " " x^(2) =y - 4 ge 0`
`rArr " "x = sqrt(y-4) ge 0`
Now for each `y in [ 4, oo[` there exist ` x in R_+` such that `f(x) = f( sqrt(y-4)) = ( sqrt(y-4))^(2) + 4 = y - 4 + 4 = y`
`therefore f `is onto.
Therefore, `f` is one-one function.
`rArr f` is invertible.
`therefore f^(-1) : [ 4, oo[ to R_+` is defined as `f^(-1) (y) = sqrt( y-4)`.
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