A function f:R-{0} to R-{3} given by f(x)= (3x+2)/x, prove that f is one one and onto.
Answers
Answer:
A=R−{3}
B=R−{1}
f:A→B
f(x)=
x−3
x−2
f(x
1
)=f(x
2
)
x
1
−3
x
1
−2
=
x
2
−3
x
2
−2
(x
2
−3)(x
1
−2)=(x
2
−2)(x
1
−3)
x
1
x
2
−3x
1
−2x
2
+6=x
1
x
2
−3x
2
−2x
1
+6
−3x
1
−2x
2
=−3x
2
−2x
1
−x
1
=−x
2
x
1
=x
2
So, f(x) is one-one
f(x)=
x−3
x−2
y=
x−3
x−2
y(x−3)=x−2
yx−3y=x−2
yx−x=3y−2
x(y−1)=3y−2
x=
(y−1)
3y−2
f(x)=
x−3
x−2
=
y−1
3y−2
−3
y−1
3y−2
−2
=
y−1
3y−2−3(y−1)
y−1
3y−2−2(y−1)
=
3y−2−3y+3
3y−2−2y+2
=
−2+3
3y−2y
=y
f(x)=y
f(x) is onto.
So f(x) is bijective and invertible
f(x)=
x−3
x−2
y=
x−3
x−2
x=
y−3
y−2
x(y−3)=y−2
xy−3x=y−2
xy−y=3x−2
y(x−1)=3x−2
y=
x−1
3x−2
f
−1
(x)=
x−1
3x−2
Answer By
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