Page 12-12
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
-1
(x)
is used to denote an inverse function. Using this notation we can write: if
y =
f(x),
then
x = f
-1
(y).
Also,
f(f
-1
(x)) = x
, and
f
-1
(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
-1
(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.
Press, simultaneously if in RPN mode,
„ò
, and change the H-View
range to read: H-View: -8 8
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.