NCERT Class 12 · Ch 1 Exercise 1.2 · Q10
NCERT Class 12 Maths · Chapter 1 Relations And Functions · Exercise 1.2 Question 10
Let A=ℝ∖{3}, B=ℝ∖{1}, and f:A→B, f(x)=(x−2)/(x−3). Is f one-one and onto? Justify.
💡 Concept Hints
- Rewrite the function as 1+1/(x−3), and check that the inverse candidate avoids the excluded input.
- Separate the one-one and onto claims. For onto, ensure each inverse candidate lies in the allowed domain.
✅ Step-by-Step Solution
Final Answer: f is bijective; inverse x=(3y−2)/(y−1).
- First f(x)=1+1/(x−3) never equals 1 for x∈A, so the given codomain is valid.
- If (a−2)/(a−3)=(b−2)/(b−3), cross-multiplication is valid because a,b≠3. Expansion gives ab−3a−2b+6=ab−3b−2a+6, so a=b.
- For y≠1, solve y(x−3)=x−2 to obtain x=(3y−2)/(y−1). Moreover x−3=1/(y−1)≠0, so x∈A.
- Substitution gives f(x)=y. Thus every element of B is reached, proving onto; with injectivity this proves bijectivity.
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