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Find the inverse of each of the followin...

Find the inverse of each of the following functions :
`f(x) = {{:(x"," -oo lt x lt 1),(x^(2)"," 1 le x le 4),(2x"," 4 lt x lt oo):}`

Text Solution

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We have
`f(x) = {{:(x, -oo lt x lt 1, -oo lt y lt 1),(x^(2), 1 le x le 2, 1 le y le 4),(2x, 2 lt x lt oo, 4 lt y lt oo):}`
the graph of f(x) is as shown in figure.
We can see from the graph that the given function is strictly increasing and hence one-one onto in `(-oo, oo)`:
Conseuqently `f^(-1)(x) = {{:(x,,,-oo lt x lt 1),(sqrt(x),,,1 le x le 4),((x)/(2),,,4 lt x lt oo):}`
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