Graphically, the inverse is a reflection of [math]f[/math] across the diagonal line [math]y=x[/math].

The [i]inverse [/i]of a function [math]f[/math], called [math]f^{-1}[/math], is the function that "undoes" [math]f[/math]. For example, the square root function [math]\sqrt{x}[/math] "undoes" the function [math]x^2[/math] (for [math]x \geq 0[/math]). Graphically, the inverse is a reflection of [math]f[/math] across the diagonal line [math]y=x[/math]. This can be thought of as simply switching the [math]x[/math] and [math]y[/math] values of each point on the graph of [math]f[/math].[br][br]Note that the inverse of a function might not itself be a function. The inverse of [math]x^2[/math] yields a parabola opening right, which fails the vertical line test. If you think about it, the square root of a positive number (such as 9) could be either positive or negative (-3 or 3). So each input to the square root rule should have two outputs. We restrict the square root rule to only positive outputs, which then makes this a function.[br][br]Drag the red dot along the function. As the red dot moves, its reflection (the black circle) will trace out points on the inverse of [math]f[/math] (in blue). Note that every for every point [math](a, b)[/math] on [math]f[/math], there is a point [math](b, a)[/math] on the inverse [math]f^{-1}[/math]. Points [math](a, b)[/math] and [math](b, a)[/math] are reflections of each other across the line [math]y=x[/math] (the "diagonal"). Click the "Clear Trace" button clear the inverse function trace.[br][br]You can enter a different function in the "[b]f(x) =[/b]" box.[br][br]A function is called [i][b]one-to-one[/b][/i] if its inverse is also a function. One input to one output ([math]f[/math] is a function) AND one output to one input ([math]f^{-1}[/math] is a function). Graphically, we can test this by using the Vertical Line Test (VLT) to determine whether [math]f[/math] is a function, and the Horizontal Line Test (HLT) to determine if [math]f^{-1}[/math] is a function. Since a horizontal line is the inverse of a vertical line, a [u]horizontal[/u] line through [math]f[/math] is equivalent to a [u]vertical[/u] line through [math]f^{-1}[/math]. Check the box "Show HLT" to show the horizontal inverse of the vertical cursor.