Classify the function as injection, surjection or bijectionf: Q → Q, defined by f(x) = x3 + 1

Asked by Sakshi | 1 year ago |  56

##### Solution :-

Given f: Q → Q, defined by f(x) = x3 + 1

Now we have to check for the given function is injection, surjection and bijection condition.

Injection test:

Let x and y be any two elements in the domain (Q), such that f(x) = f(y).

f(x) = f(y)

x+ 1 = y+ 1

x3 = y3

x = y

So, f is an injection.

Surjection test:

Let y be any element in the co-domain (Q), such that f(x) = y for some element x in Q (domain).

f(x) = y

x3+ 1 = y

x =$$3\sqrt{y-1}$$, which may not be in Q.

For example, if y= 8,

x3+ 1 =  8

x3= 7

x =$$3\sqrt{7}$$, which is not in Q.

So, f is not a surjection and f is not a bijection.

Answered by Aaryan | 1 year ago

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