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If ` f:[0,oo) to [0,1), " and " f(x)=(x)/(1+x)` then check the nature of the function. |
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Answer» Given that `f:[0,oo) to [0, oo),f(x)=(x)/(x+1)` Let `f(x_(1))=f(x_(2))` `implies (x_(1))/(x_(1)+1)=(x_(2))/(x_(2)+1)` `implies x_(1)x_(2)+x_(1)=x_(1)x_(2)+x_(2)` `implies x_(1)=x_(2)`. Thus f(x) is one-one. Now let `y=(x)/(1+x)` `implies y+yx=x` `implies x=(y)/(1-y)` As ` x ge 0, (y)/(1-y) ge 0` `implies (y)/(1-y) le 0` `implies 0 le y lt 1` or range of f(x) is `[0,1).` Thus f(x) is onto. |
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