Question

is f^-1 (x) = (4x-2)/3 inverse of f(x)=3x/4 +1/2

Answer 1

Yes.

Method 1 (compose):
f(f⁻¹(x)) = 3[(4x − 2)/3]/4 + 1/2 = (4x − 2)/4 + 1/2 = x − 1/2 + 1/2 = x.
f⁻¹(f(x)) = (4(3x/4 + 1/2) − 2)/3 = (3x + 2 − 2)/3 = x.

Method 2 (solve for inverse):
y = 3x/4 + 1/2 ⇒ y − 1/2 = 3x/4 ⇒ x = 4(y − 1/2)/3 = (4y − 2)/3.

So f⁻¹(x) = (4x − 2)/3 is indeed the inverse.