Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #225

Prove f:R ∖ {3}→R ∖ {1}, f(x)=(x−2)/(x−3), is bijective.

✅ Step-by-Step Solution

Final Answer: Bijective; f⁻¹(y)=(3y−2)/(y−1).

For x≠3, f(x)=1+1/(x−3), which never equals 1, so the stated map is well defined.
If f(u)=f(v), then the nonzero reciprocals 1/(u−3),1/(v−3) are equal, forcing u=v and proving injectivity.
For any y≠1 solve y=(x−2)/(x−3): (y−1)x=3y−2, so x=(3y−2)/(y−1).
This x obeys x−3=1/(y−1)≠0 and f(x)=y, so the map is onto R ∖ {1}.
Thus f is bijective with inverse f⁻¹(y)=(3y−2)/(y−1).

[1 mark] For x≠3, f(x)=1+1/(x−3), which never equals 1, so the stated map is well defined.
[1 mark] If f(u)=f(v), then the nonzero reciprocals 1/(u−3),1/(v−3) are equal, forcing u=v and proving injectivity.
[1 mark] For any y≠1 solve y=(x−2)/(x−3): (y−1)x=3y−2, so x=(3y−2)/(y−1).
[1 mark] This x obeys x−3=1/(y−1)≠0 and f(x)=y, so the map is onto R ∖ {1}.
[1 mark] Thus f is bijective with inverse f⁻¹(y)=(3y−2)/(y−1).

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