Exploring Inverse Functions
Prof. G. Battaly, Westchester Community College
Start with a one-to-one function, a function that passes both the vertical line test and the horizontal line test. If you exchange the x and y coordinates, that is exchange the independent and dependent variables, the result is a new function that is the inverse of the original. Here we look at the exponential function:
g(x) = 2x
Examine the function, g(x), below. Begin by moving the "b" on the slider in the upper left portion of the graph window. Click and drag the "b". (For greater precision, click on the letter then use the arrow keys on your keyboard.)
"b" is the x-coordinate of the point B on g(x).
Note the coordinates of select points on g(x), and find the corresponding inverse points. For example find point (0,1) on g(x). Then use the slider for "a" to locate the point (1,0).
Do this for "b" values of -2, -1, 0, 1, and 2. As you find the inverse points, select the point with those coordinates in the algebra window to identify the points which you have found.
Notice that the inverse points seem to be along a curve. In fact this curve is f(x) =log2 x
Click on the function f(x) in the algebra window to see how the curve fits the inverse points.

Given that g(x) = 2x and f(x) =log2 x are inverse functions,
1. If (0, 1) is a point on g(x), then the point (__, __) is on f(x).
2. If (1, 2) is a point on g(x), then the point (__, __) is on f(x).
3. If (-1, 1/2) is a point on g(x), then the point (__, __) is on f(x).
4. If (a, b) is a point on g(x), then the point (__, __) is on f(x).
5. What is the inverse function of g(x) = 10x ? f(x) = _________________
6. What is the inverse function of g(x) = ex ? f(x) = _________________
G. Battaly, WCC, 18 November 2013, Created with GeoGebra
For answers, see: http://www.battaly.com/algebra_trig/geogebra/inverse_functions