Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #188

Prove f:[0,∞)→[4,∞), f(x)=x²+4, is invertible; obtain its inverse.

✅ Step-by-Step Solution

Final Answer: f⁻¹(y)=√(y−4).

If x≥0, then x²+4≥4, so f maps into the stated codomain.
If f(u)=f(v), then u²=v² and (u−v)(u+v)=0. Nonnegative u,v give u=v (if u+v=0 both are 0).
For an arbitrary y≥4, x=√(y−4) is real and nonnegative.
This input satisfies f(x)=(√(y−4))²+4=y. Thus every codomain value has a preimage, proving surjectivity.
Consequently f is bijective and its inverse is f⁻¹:[4,∞)→[0,∞), f⁻¹(y)=√(y−4).

[1 mark] If x≥0, then x²+4≥4, so f maps into the stated codomain.
[1 mark] If f(u)=f(v), then u²=v² and (u−v)(u+v)=0. Nonnegative u,v give u=v (if u+v=0 both are 0).
[1 mark] For an arbitrary y≥4, x=√(y−4) is real and nonnegative.
[1 mark] This input satisfies f(x)=(√(y−4))²+4=y. Thus every codomain value has a preimage, proving surjectivity.
[1 mark] Consequently f is bijective and its inverse is f⁻¹:[4,∞)→[0,∞), f⁻¹(y)=√(y−4).

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