Inverse of a Function (Algebra) and Derivative of Inverse (Calculus)

Consider a function y=f(x) and its inverse relation x=f(y). As implied by the notation, the input and output values are swapped between the point with coordinates (x, y) on the graph of function y=f(x) and the corresponding point with coordinates (y, x) on the graph of relation x=f(y). Drag the slider in the construction above all the way from left to right to display the graphs of both the given function and its inverse relation. (Make sure both checkboxes under the slider are toggled on.) Visually, how does the graph of function y=f(x) relate to the graph of inverse relation x=f(y)?

In the preceding text, care was taken to refer to x=f(y) as an "inverse relation" rather than an "inverse function." Why do you think that is?

What conditions of a function y=f(x) would yield an inverse x=f(y) relation that also satisfies the definition of "function"?

For the given default function , the inverse relation is not a function. Toggle on the "Restrict range of inverse" checkbox. Drag the two points that appear along the right edge of the graphics view up/down until the inverse graph passes the vertical line test (i.e. no two points on the graph share the same x value). Don't trim the graph back any more than is necessary to meet this condition. What is the range of the resulting inverse function ?

Toggle on the "Tangent lines" checkbox. Drag point A back and forth along the graph of y=f(x). You also see the corresponding point B on the graph of y=f^-1(x). Comparing the lines tangent to the graphs of y=f(x) at point A and y=f^-1(x) at point B: Do the slopes of the two lines have the same sign or opposite signs? Is this always true for all points on all functions?

Drag point A to several different locations along the graph of y=f(x), making sure to keep point B on the graph of y=f^-1(x). Using the displayed grid on the x-plane, estimate and compare the slopes of each tangent line to each other. Speculate: What is the relationship between the slopes of the corresponding lines tangent to the graph of y=f(x) at point A and tangent to the graph of y=f^-1(x) at point B?

Consider the definition of "slope" and the fact that a point A with coordinates (x,y) on the graph of y=f(x) corresponds to a point B with coordinates (y,x) on the graph of relation x=f(y). Why does it make intuitive sense that the slopes of the tangent lines have the relationship established in the previous question?