Finding an Inverse Algebraically

Finding an inverse algebraically means writing y = f(x), swapping x and y, solving for the new y, and confirming with the composition test.

Invert f(x) = 2x + 3. Write y = 2x + 3. Swap the letters — the reversal itself: x = 2y + 3. Solve for y: x − 3 = 2y, so y = (x − 3)/2. Name it: f⁻¹(x) = (x − 3)/2. Certify: f(f⁻¹(x)) = 2 · (x − 3)/2 + 3 = x, and the other direction runs to x as well. The swap is the whole idea in one move — inputs and outputs trading seats — and the solving that follows is Video-133 machinery with a new letter in the chair.