ANSWER
[tex]k = \pm \frac{1}{4} [/tex]
EXPLANATION
The given functions are
[tex]f(x) =( \frac{1}{2} )^{x} [/tex]
and
[tex]g(x) = {k}^{ - x + 6} [/tex]
We were given that, the two functions are equal only when
[tex]x = 4[/tex]
This implies that,
[tex]f(4) = g(4)[/tex]
This will give us the equation,
[tex]( \frac{1}{2} )^{4} = {k}^{ - 4 + 6} [/tex]
We simplify the exponent on k to obtain,
[tex]( \frac{1}{2} )^{4} = {k}^{ 2} [/tex]
This implies that,
[tex] \frac{1}{16} = {k}^{ 2} [/tex]
We take the square root of both sides to get,
[tex] \pm \: \sqrt{ \frac{1}{16} } = k[/tex]
[tex]k = \pm \frac{1}{4} [/tex]
Both values satisfies the equation for
[tex]x = 4[/tex]
