Test: Inverse Functions - Grade 12 MCQ

Inverse of a function is the mirror image of function in line y = x.

Compostion of a function with its inverse gives x.

It is not a one one function hence Inverse does not exist.

y = 3x-5, x = (y+5)/3, f ^-1(x) = (x+5)/3.

Inverse associate each element in B with corresponding element in A.

On giving inverse, image the function returns preimage thus f^-1 (y) = x.

Inverse exist only for those functions which are one one and onto.

Let sin⁻¹ x = θ , 
Then sinθ = x
⇒ cscθ = 1/x
⇒ csc² θ = 1/x²
⇒1+ cot² θ = 1/x²
⇒ cot = √(1 – x²) /x

Substituting x=tanA
tan^-1(sec A-1)/tanA = tan^-1(1-cosA)/sinA
= tan^-1(tan(A/2)).... Using cos A = 1-2sin² A/2 
4°=A/2
tan ⁻¹(x)=8°
⇒ A= 8°
x =tan (8°)

The equation is cos^-1 x + cos^-1 y + cos^-1 z = 3π
This means cos^-1 x = π, cos^-1 y = π and cos^-1 z = π
This will be only possible when it is in maxima.
As, cos^-1 x = π so, x = cos^-1 π = -1 similarly, y = z = -1
Therefore, x + y + z = -1 -1 -1
So, x + y + z = -3.