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Mathematics
Mathematics
Problem
Let
g
:
R
→
R
g:\mathbb R\to\mathbb R
g
:
R
→
R
be given by
g
(
x
)
=
2
x
+
5
3
g(x)=\dfrac{2x+5}{3}
g
(
x
)
=
3
2
x
+
5
, and let
h
h
h
denote the inverse of
g
g
g
. (i) Prove that
h
(
g
(
x
)
)
=
x
h\bigl(g(x)\bigr)=x
h
(
g
(
x
)
)
=
x
for every real
x
x
x
. (ii) Simplify the expression
g
(
h
(
g
(
h
(
x
)
)
)
)
g\bigl(h\bigl(g\bigl(h(x)\bigr)\bigr)\bigr)
g
(
h
(
g
(
h
(
x
)
)
)
)
, showing each algebraic step clearly. (iii) Hence find the numerical value of
g
(
h
(
g
(
h
(
11
)
)
)
)
g\bigl(h\bigl(g\bigl(h(11)\bigr)\bigr)\bigr)
g
(
h
(
g
(
h
(
11
)
)
)
)
.
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