Question

A function f : R → R is defined as f(x) = 3 − || Show that f(x) is not surjective.

Answer 1

Answer:

one−one test for f:

Let x and y be two elements of domain (R), such that

⇒ f(x)=f(y)

⇒ x

3

+4=y

3

+4

⇒ x

3

=y

3

⇒ x=y

∴ f is one-one.

onto test for f:

Lety be in the co-domain (R), such that,

⇒ f(x)=y

⇒ x

2

+4=y

⇒ x=

3

y−4

∈R (Domain)

∴ f is onto.

Since, f is one-one and onto then it is bijective.

∴ f is invertible.

Find f

−1

:

Let f

−1

(x)=y ----- ( 1 )

⇒ x=f(y)

⇒ x=y

3

+4

⇒ x−4=y

3

⇒ y=

3

x−4

From ( 1 ),

⇒ f

−1

(x)=

3

x−4

⇒ f

−1

(3)=

3

3−4

=

3

−1

=−1

Step-by-step explanation:

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