Evaluate $(h h)\left(\frac{7}{5}\right)$ given that $h(x)=\frac{1}{5 x+2}$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. $( hh )\left(\frac{7}{5}\right)=$ $\square$ (Simplify your answer.) B. The answer is undefined.

First, find h ( f r a c 7 5 ) by substituting f r a c 7 5 into h ( x ) = f r a c 1 5 x + 2 , which gives h ( f r a c 7 5 ) = f r a c 1 9 .
Then, find h ( h ( f r a c 7 5 )) = h ( f r a c 1 9 ) by substituting f r a c 1 9 into h ( x ) , which gives h ( f r a c 1 9 ) = f r a c 1 5 ( f r a c 1 9 ) + 2 = f r a c 9 23 .
Therefore, ( hh ) ( f r a c 7 5 ) = f r a c 9 23 .
The final answer is f r a c 9 23 ​ .

Explanation

Understanding the problem We are given the function h ( x ) = f r a c 1 5 x + 2 and we need to evaluate ( hh ) ( f r a c 7 5 ) , which means we need to find h ( h ( f r a c 7 5 )) .

Evaluating h(7/5) First, we evaluate h ( f r a c 7 5 ) by substituting x = f r a c 7 5 into the expression for h ( x ) : h ( f r a c 7 5 ) = f r a c 1 5 ( f r a c 7 5 ) + 2 = f r a c 1 7 + 2 = f r a c 1 9 So, h ( f r a c 7 5 ) = f r a c 1 9 .

Evaluating h(h(7/5)) Next, we evaluate h ( h ( f r a c 7 5 )) = h ( f r a c 1 9 ) by substituting x = f r a c 1 9 into the expression for h ( x ) : h ( f r a c 1 9 ) = f r a c 1 5 ( f r a c 1 9 ) + 2 = f r a c 1 f r a c 5 9 + 2 = f r a c 1 f r a c 5 9 + f r a c 18 9 = f r a c 1 f r a c 23 9 = f r a c 9 23 Therefore, ( hh ) ( f r a c 7 5 ) = f r a c 9 23 .

Final Answer Thus, ( hh ) ( f r a c 7 5 ) = f r a c 9 23 .


Examples
Composite functions are used in various real-world scenarios. For example, consider a store that offers a discount of 10% on all items and then applies a sales tax of 5%. If x is the original price, the discounted price is f ( x ) = 0.9 x , and the price after tax is g ( x ) = 1.05 x . The final price after both discount and tax is g ( f ( x )) , which is a composite function. Understanding composite functions helps in calculating the final price accurately.

Answered by GinnyAnswer | 2025-07-03

Evaluating ( hh ) ( 5 7 ​ ) gives 23 9 ​ after calculating both h ( 5 7 ​ ) and h ( h ( 5 7 ​ )) . The final answer is 23 9 ​ .
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Answered by Anonymous | 2025-07-04