Class 12 Relations And Functions · Case Study4 Markstypes_of_functions
Class 12 Maths · Relations And Functions · Case Study #198
Rows in a seating layout motivate the straight line y=x−4. Let L be the set of all lines in the plane; “same direction” includes coincident lines.
Part (1) 1 Marks
On L, define lRm when l and m have the same direction. Is R an equivalence relation?
Answer: Yes.
- Every line has its own direction; equal directions are symmetric and equality of directions is transitive.
- R is reflexive, symmetric and transitive, hence an equivalence relation.
Part (2) 1 Marks
On L, define lSm when l is perpendicular to m. State the properties.
Answer: Symmetric, not reflexive or transitive.
- Perpendicularity is symmetric, but no line is perpendicular to itself; l⊥m and m⊥l do not imply l⊥l.
- Therefore S is symmetric but neither reflexive nor transitive.
Part (3) 2 Marks
Prove f:R→R, f(x)=x−4, is bijective.
Answer: Bijective; f⁻¹(y)=y+4.
- If f(u)=f(v), then u−4=v−4 and u=v.
- For any y∈R choose x=y+4; then f(x)=y, proving onto.
- Hence f is bijective and its inverse is y↦y+4.
Part (4) 2 Marks
Alternatively, find the range of g:R→R, g(x)=x+4.
Answer: R
- Let y be any real target and solve y=x+4.
- The input x=y−4 is real and satisfies g(x)=y, so every real target is attained.
- Thus the range is R.