Example INV.REFL.0   Suppose $f(x) = -2x + 1$. Verify that $g(x) = -\frac {x-1} 2$ is the inverse function for $f$..
(i)$(f \circ g)(x) = f(g(x)) =f(-\frac {x-1} 2) = -2*(-\frac {x-1} 2) + 1 = (x-1) +1= x$.
(ii)$(g \circ f)(x) = g(f(x)) =g(- 2x + 1) =  -\frac {(-2x + 1)-1} 2 = x$.

We check (i) visually both with the graph and the Mapping Diagram.
Which do you find more convincing visually?

Select the point $x$ in the mapping diagram to move the point by mouse or use the scroll up or down key  to move by increments.