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GeoGebraGeoGebra Classroom

Graphs of Function Inverses & One-to-One Functions

Graphically, the inverse is a reflection of across the diagonal line .
The inverse of a function , called , is the function that "undoes" . For example, the square root function "undoes" the function (for ). Graphically, the inverse is a reflection of across the diagonal line . This can be thought of as simply switching the and values of each point on the graph of . Note that the inverse of a function might not itself be a function. The inverse of 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. 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 (in blue). Note that every for every point on , there is a point on the inverse . Points and are reflections of each other across the line (the "diagonal"). Click the "Clear Trace" button clear the inverse function trace. You can enter a different function in the "f(x) =" box. A function is called one-to-one if its inverse is also a function. One input to one output ( is a function) AND one output to one input ( is a function). Graphically, we can test this by using the Vertical Line Test (VLT) to determine whether is a function, and the Horizontal Line Test (HLT) to determine if is a function. Since a horizontal line is the inverse of a vertical line, a horizontal line through is equivalent to a vertical line through . Check the box "Show HLT" to show the horizontal inverse of the vertical cursor.