Show that the function f : R R : f (x) = x4 is neither one-one nor onto.

To prove: function is neither one-one nor onto


Given: f : R R : f (x) = x4


We have,


f(x) = x4


For, f(x1) = f(x2)


x14 = x24


(x14 - x24) = 0


(x12 - x22) (x12 + x22) = 0


(x1 - x2) (x1 + x2) (x12 + x22) = 0


x1 = x2 or, x1 = -x2 or, x12 = -x22


We are getting more than one value of x1 (no unique image)


f(x) is not one-one


f(x) = x4


Let f(x) = y such that


y = x4



If y = -2, as


Then x will be undefined as we can’t place the negative value under the square root


Hence f(x) is not onto


Hence Proved


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