Class 12 Relations And Functions · Very Long Answer5 Markstypes_of_functions

Class 12 Maths · Relations And Functions · Very Long Answer #187

Prove f:[0,∞)→[−9,∞), f(x)=5x²+6x−9, is invertible and find its inverse.

✅ Step-by-Step Solution

Final Answer: f⁻¹(y)=(−3+√(5y+54))/5, y≥−9.

For x≥0, f(x)+9=x(5x+6)≥0, so outputs belong to the specified codomain.
If f(u)=f(v), then (u−v)(5(u+v)+6)=0. Since u,v≥0 the second factor is positive; therefore u=v.
Solve 5x²+6x−9=y to obtain x=(−3±√(5y+54))/5.
For y≥−9, √(5y+54)≥3, so the plus root is nonnegative and the minus root is negative. The plus root is an allowed preimage and substitution satisfies the equation.
Thus f is bijective and invertible, with f⁻¹(y)=(−3+√(5y+54))/5 for y≥−9.

[1 mark] For x≥0, f(x)+9=x(5x+6)≥0, so outputs belong to the specified codomain.
[1 mark] If f(u)=f(v), then (u−v)(5(u+v)+6)=0. Since u,v≥0 the second factor is positive; therefore u=v.
[1 mark] Solve 5x²+6x−9=y to obtain x=(−3±√(5y+54))/5.
[1 mark] For y≥−9, √(5y+54)≥3, so the plus root is nonnegative and the minus root is negative. The plus root is an allowed preimage and substitution satisfies the equation.
[1 mark] Thus f is bijective and invertible, with f⁻¹(y)=(−3+√(5y+54))/5 for y≥−9.

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