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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]

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