If $h(x)=5+x$ and $k(x)=\frac{1}{x}$, which expression is equivalent to $(k \circ h)(x) ?
$\frac{(5+x)}{x}$
$\frac{1}{(5+x)}$
$5+\left(\frac{1}{x}\right)$
$5+(5+x)$
Answer (1)
Find h ( x ) : h ( x ) = 5 + x .
Substitute h ( x ) into k ( x ) : k ( h ( x )) = k ( 5 + x ) .
Evaluate k ( 5 + x ) : k ( 5 + x ) = 5 + x 1 .
The expression equivalent to ( k c i rc h ) ( x ) is 5 + x 1 .
Explanation
Understanding the Problem We are given two functions, h ( x ) = 5 + x and k ( x ) = x 1 . We want to find the composite function ( k c i rc h ) ( x ) , which means we need to evaluate k ( h ( x )) .
Finding h(x) First, we find h ( x ) , which is given as h ( x ) = 5 + x .
Substituting h(x) into k(x) Next, we substitute h ( x ) into k ( x ) . So we have k ( h ( x )) = k ( 5 + x ) . Since k ( x ) = x 1 , we replace x with ( 5 + x ) in the expression for k ( x ) . This gives us k ( 5 + x ) = 5 + x 1 .
Final Result Therefore, ( k c i rc h ) ( x ) = 5 + x 1 .
Examples
In manufacturing, if h ( x ) represents the number of units produced by a machine in x hours, and k ( x ) represents the cost per unit when producing x units, then ( k c i rc h ) ( x ) gives the cost per unit as a function of the number of hours the machine operates. For example, if h ( x ) = 5 + x (machine produces 5 units initially plus x units per hour) and k ( x ) = x 1 (cost per unit decreases as more units are produced), then ( k c i rc h ) ( x ) = 5 + x 1 represents the cost per unit as a function of the number of hours the machine operates. This helps in optimizing production time to minimize cost per unit.
