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Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1978 AMC 12/AHSME
17
17
Part of
1978 AMC 12/AHSME
Problems
(1)
Function such that f(x^2+1)^sqrt(x)=k
Source: 1978 AHSME Problem 17
6/6/2014
If
k
k
k
is a positive number and
f
f
f
is a function such that, for every positive number
x
x
x
,
[
f
(
x
2
+
1
)
]
x
=
k
;
\left[f(x^2+1)\right]^{\sqrt{x}}=k;
[
f
(
x
2
+
1
)
]
x
=
k
;
then, for every positive number
y
y
y
,
[
f
(
9
+
y
2
y
2
)
]
12
y
\left[f(\frac{9+y^2}{y^2})\right]^{\sqrt{\frac{12}{y}}}
[
f
(
y
2
9
+
y
2
)
]
y
12
is equal to
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
k
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
2
k
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
k
k
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
k
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
y
k
<span class='latex-bold'>(A) </span>\sqrt{k}\qquad<span class='latex-bold'>(B) </span>2k\qquad<span class='latex-bold'>(C) </span>k\sqrt{k}\qquad<span class='latex-bold'>(D) </span>k^2\qquad <span class='latex-bold'>(E) </span>y\sqrt{k}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
k
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
2
k
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
k
k
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
k
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
y
k
function
AMC