Inverse functions and their graphs
Let y = f(x), if we can find a function y = g(x), such that, g(f(x)) = x, then we
say that g(x) is the inverse function of f(x). Typically, the notation g(x) = f
is used to denote an inverse function. Using this notation we can write: if y =
f(x), then x = f
(y). Also, f(f
(x)) = x, and f
(f(x)) = x.
As indicated earlier, the ln(x) and exp(x) functions are inverse of each other,
i.e., ln(exp(x)) = x, and exp(ln(x)) = x. This can be verified in the calculator
by typing and evaluating the following expressions in the Equation Writer:
LN(EXP(X)) and EXP(LN(X)). They should both evaluate to X.
When a function f(x) and its inverse f
(x) are plotted simultaneously in the
same set of axes, their graphs are reflections of each other about the line y =
x. Let's check this fact with the calculator for the functions LN(X) and EXP(X)
by following this procedure:
Press, simultaneously if in RPN mode, „ñ. The function Y1(X) = EXP(X)
should be available in the PLOT - FUNCTION window from the previous
exercise. Press @@ADD@! , and type the function Y2(X) = LN(X). Also, load
the function Y3(X) = X. Press L@@@OK@@@ to return to normal calculator display.
„ò, and change the H-View
Press, simultaneously if in RPN mode,
range to read:
Press @AUTO to generate the vertical range. Press @ERASE @DRAW to produce the
graph of y = ln(x), y = exp(x), and y =x, simultaneously if in RPN mode.
You will notice that only the graph of y = exp(x) is clearly visible. Something
went wrong with the @AUTO selection of the vertical range. What happens is
that, when you press @AUTO in the PLOT FUNCTION – WINDOW screen, the
calculator produces the vertical range corresponding to the first function in the
list of functions to be plotted. Which, in this case, happens to be Y1(X) =
EXP(X). We will have to enter the vertical range ourselves in order to display
the other two functions in the same plot.