Function Inverses
The EPIC screen below shows the beginning of the choice for drawing the inverse relation corresponding to the function F(x) = x cos(x), a <= x <= b, with a = -5 and b = 5. Normally, "Superimpose" is the desired choice. Furthermore, if the viewing window is square, the line y = x will be drawn along with the inverse relation.
The next EPIC screen shows the graph of the inverse relation. The black curve is the original function, and the red curve is the inverse relation. As mentioned above, the graph of the line y = x has also been drawn. Clearly this curve fails the vertical line test for functions! This function does not have an inverse FUNCTION on the chosen interval [-5,5].
Next, we used the "Roots" option on the 2D Graphs main menu to find the critical points of F(x) -- that is, we had EPIC (numerically) solve F'(x) = 0 -- obtaining four values: -3.425618, -0.8603336, 0.8603336 and 3.425618. (These values are automatically placed on a "Recent Data" list, so when "Parameters" was chosen, the F2 key was pressed to access this list, and we simply chose the values we wanted. There was no need to copy the values down, and re-enter them.) Now between these values, F -1(x) IS a function. The next screen shows the restriction of F to the interval [-3.425618, -0.8603336], and its inverse function on that interval.
Finally, the restriction of F to [-0.8603336, 0.8603336] also has an inverse; we have reset the viewing window to [-1,1] by [-1,1]. As noted by the color of F in the heading of the screen, the graph of F is in blue; the graph of F-1 is sort of pink. Notice how the slope of F-1(x) becomes infinite as we approach the endpoints of the interval (where F'(x) = 0). Very Nice!
