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Ireland National Math Olympiad
2021 Irish Math Olympiad
5
5
Part of
2021 Irish Math Olympiad
Problems
(1)
g(g(x)) = 3x/(x + 3) for x>=0, g(x) = (x + 1)/2 for 2<=x<=3
Source: 2021 Irish Mathematical Olympiad P5
5/30/2021
The function
g
:
[
0
,
∞
)
→
[
0
,
∞
)
g : [0, \infty) \to [0, \infty)
g
:
[
0
,
∞
)
→
[
0
,
∞
)
satisfies the functional equation:
g
(
g
(
x
)
)
=
3
x
x
+
3
g(g(x)) = \frac{3x}{x + 3}
g
(
g
(
x
))
=
x
+
3
3
x
, for all
x
≥
0
x \ge 0
x
≥
0
. You are also told that for
2
≤
x
≤
3
2 \le x \le 3
2
≤
x
≤
3
:
g
(
x
)
=
x
+
1
2
g(x) = \frac{x + 1}{2}
g
(
x
)
=
2
x
+
1
. (a) Find
g
(
2021
)
g(2021)
g
(
2021
)
. (b) Find
g
(
1
/
2021
)
g(1/2021)
g
(
1/2021
)
.
algebra
functional
functional equation