Which of the following functions form Z into Z bijections? (a) f(x) = (b)) - 2 (c) f(x) = 2x + 1 malosole
Answers
Step-by-step explanation:
f(x)=x
3
one−one test:
Let x
1
and x
2
be the elements in the domain such that,
f(x
1
)=f(x
2
)
⇒ x
1
3
=x
2
3
⇒ x
1
=x
2
∴ f is one-one.
onto test :
Let y be an element in the co-domain (Z), such that,
f(x)=y
⇒ x
3
=y
Consider y=3
⇒ x
3
=3
⇒ x=
3
3
∈
/
Z
∴ f is not onto.
∴ f is not bijection.
(B) f(x)=x+2
one−one test:
Let x
1
and x
2
be the elements in the domain such that,
f(x
1
)=f(x
2
)
⇒ x
1
+2=x
2
+2
⇒ X
1
=x
2
∴ f is one-one.
onto test :
Let y be an element in the co-domain (Z), such that,
f(x)=y
⇒ x+2=y
⇒ x=y−2∈Z(Domain)
∴ f is onto
∴ f is bijection.
(C) f(x)=2x+1
one−one test:
Let x
1
and x
2
be the elements in the domain such that,
f(x
1
)=f(x
2
)
⇒ 2x
1
+1=2x
2
+1
⇒ 2x
1
=2x
2
⇒ x
1
=x
2
∴ f is one-one.
onto test :
Let y be an element in the co-domain (Z), such that,
f(x)=y
⇒ 2x+1=y
Consider y=4
⇒ 2x+1=4
⇒ 2x=3
⇒ x=
2
3
∈
/
Z
∴ f is not onto.
∴ f is not bijection.
(D) f(x)=x
2
+x
⇒ f(0)=(0)
2
+0=0
⇒ f(−1)=(−1)
2
+(−1)=1−1=0
We can see that 0 and −1 have the same image.
∴ f is not on-one.
∴ f is not bijection.