Information Theory, Inference, and Learning ... - MAELabs UCSD

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4.8: Solutions 89
Solution to exercise 4.20 (p.86). The function f(x) has inverse function
Note
g(y) = y 1/y . (4.54)
log g(y) = 1/y log y. (4.55)
I obtained a tentative graph of f(x) by plotting g(y) with y along the vertical
axis and g(y) along the horizontal axis. The resulting graph suggests that
f(x) is single valued for x ∈ (0, 1), and looks surprisingly well-behaved and
ordinary; for x ∈ (1, e 1/e ), f(x) is two-valued. f( √ 2) is equal both to 2 and
4. For x > e 1/e (which is about 1.44), f(x) is infinite. However, it might be
argued that this approach to sketching f(x) is only partly valid, if we define f
as the limit of the sequence of functions x, xx , xxx, . . .; this sequence does not
have a limit for 0 ≤ x ≤ (1/e) e 0.07 on account of a pitchfork bifurcation
at x = (1/e) e ; and for x ∈ (1, e 1/e ), the sequence’s limit is single-valued – the
lower of the two values sketched in the figure.
50
40
30
20
10
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
5
4
3
2
1
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4
0.5
0.4
0.3
0.2
0.1
0
0 0.2
Figure 4.15. f(x) = x xxxx···
, shown
at three different scales.