Let's determine which expression is equivalent to [tex]\((k \circ h)(x)\)[/tex] given that [tex]\( h(x) = 5 + x \)[/tex] and [tex]\( k(x) = \frac{1}{x} \)[/tex].
The composition of functions [tex]\( (k \circ h)(x) \)[/tex] means we first apply [tex]\( h(x) \)[/tex] and then apply [tex]\( k \)[/tex] to the result of [tex]\( h(x) \)[/tex].
Step-by-Step Solution:
1. Evaluate [tex]\( h(x) \)[/tex]:
[tex]\[
h(x) = 5 + x
\][/tex]
2. Substitute [tex]\( h(x) \)[/tex] into [tex]\( k \)[/tex]:
Since [tex]\( k(x) = \frac{1}{x} \)[/tex], we need to find [tex]\( k(h(x)) \)[/tex].
[tex]\[
k(h(x)) = k(5 + x)
\][/tex]
3. Apply the function [tex]\( k \)[/tex] to [tex]\( 5 + x \)[/tex]:
Using the definition of [tex]\( k(x) \)[/tex],
[tex]\[
k(5 + x) = \frac{1}{5 + x}
\][/tex]
So, the expression [tex]\( (k \circ h)(x) \)[/tex] simplifies to:
[tex]\[
(k \circ h)(x) = \frac{1}{5 + x}
\][/tex]
Therefore, the expression equivalent to [tex]\( (k \circ h)(x) \)[/tex] is:
[tex]\[
\boxed{\frac{1}{5 + x}}
\][/tex]
