If $h(x)=5+x$ and $k(x)=\frac{1}{x}$, which expression is equivalent to $(k \circ h)(x) ?
A. $\frac{(5+x)}{x}$
B. $\frac{1}{(5+x)}$
C. $5+\left(\frac{1}{x}\right)$
D. $5+(5+x)$
Answer (1)
Substitute h ( x ) into k ( x ) to find ( k ∘ h ) ( x ) .
Given h ( x ) = 5 + x and k ( x ) = x 1 , calculate k ( h ( x )) = k ( 5 + x ) .
Evaluate k ( 5 + x ) = 5 + x 1 .
The equivalent expression is 5 + x 1 .
Explanation
Understanding the Problem We are given two functions, h ( x ) = 5 + x and k ( x ) = x 1 . We need to find the expression that is equivalent to the composition of these functions, specifically ( k ∘ h ) ( x ) . This means we need to evaluate k ( h ( x )) .
Composition of Functions To find ( k ∘ h ) ( x ) , we need to substitute h ( x ) into the function k ( x ) . In other words, we replace the x in k ( x ) with the entire expression for h ( x ) . Since h ( x ) = 5 + x and k ( x ) = x 1 , we have: k ( h ( x )) = k ( 5 + x ) = 5 + x 1
Finding the Equivalent Expression Now we compare our result, 5 + x 1 , with the given options to find the equivalent expression. The options are:
x ( 5 + x )
( 5 + x ) 1
5 + ( x 1 )
5 + ( 5 + x )
Our result, 5 + x 1 , matches option 2.
Final Answer Therefore, the expression equivalent to ( k ∘ h ) ( x ) is 5 + x 1 .
Examples
In manufacturing, if h ( x ) represents the number of units produced by a machine and k ( x ) represents the cost per unit, then ( k ∘ h ) ( x ) gives the cost per unit based on the number of units produced. For example, if producing x units results in 5 + x units due to efficiency gains, and the cost per unit is the inverse of the number of units, then ( k ∘ h ) ( x ) = 5 + x 1 gives the cost per unit considering the efficiency gains.
