Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #192

Prove f:R ∖ {−4/3}→R ∖ {4/3}, f(x)=(4x+3)/(3x+4), is bijective and find its inverse.

✅ Step-by-Step Solution

Final Answer: f⁻¹(y)=(3−4y)/(3y−4).

The domain excludes the zero denominator. The equation f(x)=4/3 would imply 12x+9=12x+16, impossible; hence the codomain is respected.
If f(u)=f(v), cross multiplication simplifies to 7(u−v)=0, so u=v and f is injective.
For y≠4/3 solve y(3x+4)=4x+3 to get x=(3−4y)/(3y−4).
For this input x+4/3=−7/[3(3y−4)]≠0, so it is allowed; substitution in f gives y. Therefore f is onto.
Thus f is bijective with f⁻¹(y)=(3−4y)/(3y−4) on R ∖ {4/3}.

[1 mark] The domain excludes the zero denominator. The equation f(x)=4/3 would imply 12x+9=12x+16, impossible; hence the codomain is respected.
[1 mark] If f(u)=f(v), cross multiplication simplifies to 7(u−v)=0, so u=v and f is injective.
[1 mark] For y≠4/3 solve y(3x+4)=4x+3 to get x=(3−4y)/(3y−4).
[1 mark] For this input x+4/3=−7/[3(3y−4)]≠0, so it is allowed; substitution in f gives y. Therefore f is onto.
[1 mark] Thus f is bijective with f⁻¹(y)=(3−4y)/(3y−4) on R ∖ {4/3}.

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