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5
2016 Algebra #5
2016 Algebra #5
Source:
August 6, 2022
2016
Algebra Test
Problem Statement
Let
f
(
x
)
f(x)
f
(
x
)
be a real valued function. Recall that if the inverse function
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
exists, then
f
−
1
(
x
)
f^{-1}(x)
f
−
1
(
x
)
satisfies
f
(
f
−
1
(
x
)
)
=
f
−
1
(
f
(
x
)
)
=
x
f(f^{-1}(x))=f^{-1}(f(x))=x
f
(
f
−
1
(
x
))
=
f
−
1
(
f
(
x
))
=
x
. Given that the inverse of the function
f
(
x
)
=
x
3
−
12
x
2
+
48
x
−
60
f(x)=x^3-12x^2+48x-60
f
(
x
)
=
x
3
−
12
x
2
+
48
x
−
60
exists, find all real
a
a
a
that satisfy
f
(
a
)
=
f
−
1
(
a
)
f(a)=f^{-1}(a)
f
(
a
)
=
f
−
1
(
a
)
.
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