a function f is defined by f(x) = 2/x + k. Given that f^2(2) = 2/3f(1), calculate the possible values of kThis is A composite functions question...

We'll solve this problem considering the given constraint
as it follows:

[f(2)]^2 =
(2/3)*f(1)

To calculate [f(2)]^2, w'ell have to calculate
first [f(x)]^2.

[f(x)]^2 = (2/x +
k)^2

We'll expand the
square:

[f(x)]^2 = (2/x)^2 + 4k/x +
k^2

We'll calculate
[f(2)]^2:

[f(2)]^2 = (2/2)^2 + 4k/2 +
k^2

[f(2)]^2 = 1 + 2k + k^2
(1)

(2/3)*f(1) = (2/3)*(2 + k)
(2)

We'll equate (1) and
(2):

1 + 2k + k^2 = (2/3)*(2 +
k)

We'll recognize that 1 + 2k + k^2 is a perfect
square;

1 + 2k + k^2 = (1 +
k)^2

Let (1+k)^2 = t^2 => k + 2 = t +
1

t^2 - 2t/3 - 2/3 = 0

3t^2 -
2t - 2 = 0

t1 = [2 + sqrt(4 +
24)]/6

t1 = (2+2sqrt7)/6

t1 =
(1+sqrt7)/3

t2 =
(1-sqrt7)/3

But k+ 1 = t1 => k = t1 - 1 => k1
= (-2+sqrt7)/3

But k+ 1 = t2 => k = t2 - 1 =>
k2 = (-2-sqrt7)/3

The possible values of k
are: {(-2-sqrt7)/3 ; (-2+sqrt7)/3}.