Evaluate $(h h)\left(\frac{2}{5}\right)$ given that $h(x)=\frac{1}{5 x+1}$.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A. (hh) $\left(\frac{2}{5}\right)=$ $\square$ (Simplify your answer.)
B. The answer is undefined.
Answer (2)
First, find h ( 5 2 ) by substituting 5 2 into h ( x ) , which gives 3 1 .
Then, find h ( 3 1 ) by substituting 3 1 into h ( x ) , which gives 8 3 .
Therefore, ( hh ) ( 5 2 ) = 8 3 .
The final answer is 8 3 .
Explanation
Calculate h(2/5) First, we need to find h ( 5 2 ) . We substitute x = 5 2 into the expression for h ( x ) : h ( 5 2 ) = 5 ( 5 2 ) + 1 1 h ( 5 2 ) = 2 + 1 1 = 3 1
Calculate h(1/3) Next, we need to find h ( h ( 5 2 )) , which is h ( 3 1 ) . We substitute x = 3 1 into the expression for h ( x ) : h ( 3 1 ) = 5 ( 3 1 ) + 1 1 h ( 3 1 ) = 3 5 + 1 1 = 3 5 + 3 3 1 = 3 8 1 = 8 3
Final Answer Therefore, ( hh ) ( 5 2 ) = h ( h ( 5 2 )) = 8 3 .
Examples
Composite functions are used in real life to model situations where one function depends on another. For example, the cost of producing items depends on the number of items produced, and the number of items sold depends on the price. Combining these functions allows businesses to analyze how production costs affect sales.
To find ( hh ) ( 5 2 ) , we first calculate h ( 5 2 ) which equals 3 1 , and then we evaluate h ( 3 1 ) , resulting in 8 3 . Thus, the final answer is 8 3 .
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