What is the value of f^{-1)(6)

Start at 6 on the y axis. Draw an arrow so that it horizontally moves to the blue curve. Then draw an arrow straight down to vertically drop to x = 4
See the diagram below.
Follow this process in reverse to see that x = 4 leads straight up to the blue curve, and then to the left to land on y = 6. Keep in mind that the inverse reverses the input and output process, which is why I started with y instead of x.