Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #193

For f:R ∖ {−4/3}→R, f(x)=4x/(3x+4), prove it is one-one but not onto. Find its range.

✅ Step-by-Step Solution

Final Answer: Injective, not onto R; range R ∖ {4/3}.

The denominator vanishes precisely at −4/3; this value must be excluded from the domain.
If 4u/(3u+4)=4v/(3v+4), then 12uv+16u=12uv+16v, giving u=v; f is injective.
The equation f(x)=4/3 would require 12x=12x+16 and has no solution, so f is not onto R.
For y≠4/3, x=4y/(4−3y) gives x+4/3=16/[3(4−3y)]≠0 and f(x)=y, so every other real value is attained.
The function is one-one but not onto R; its range is exactly R ∖ {4/3}.

[1 mark] The denominator vanishes precisely at −4/3; this value must be excluded from the domain.
[1 mark] If 4u/(3u+4)=4v/(3v+4), then 12uv+16u=12uv+16v, giving u=v; f is injective.
[1 mark] The equation f(x)=4/3 would require 12x=12x+16 and has no solution, so f is not onto R.
[1 mark] For y≠4/3, x=4y/(4−3y) gives x+4/3=16/[3(4−3y)]≠0 and f(x)=y, so every other real value is attained.
[1 mark] The function is one-one but not onto R; its range is exactly R ∖ {4/3}.

Practise interactive questions, flashcards, visual mind maps, and timed board tests for Relations And Functions.