Evaluate $(h h)\left(\frac{7}{5}\right)$ given that $h(x)=\frac{1}{5 x+2}$

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 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 )) . In other words, we need to apply the function h to the value f r a c 7 5 , and then apply the function h again to the result.

Evaluating h(7/5) First, let's evaluate h ( f r a c 7 5 ) . We substitute 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 h ( f r a c 7 5 ) = f r a c 1 7 + 2 h ( f r a c 7 5 ) = f r a c 1 9

Evaluating h(1/9) Now, we need to evaluate h ( h ( f r a c 7 5 )) = h ( f r a c 1 9 ) . We substitute 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 h ( f r a c 1 9 ) = f r a c 1 f r a c 5 9 + 2 h ( f r a c 1 9 ) = f r a c 1 f r a c 5 + 18 9 h ( f r a c 1 9 ) = f r a c 1 f r a c 23 9 h ( f r a c 1 9 ) = f r a c 9 23

Final Answer Therefore, ( hh ) ( f r a c 7 5 ) = h ( h ( f r a c 7 5 )) = h ( f r a c 1 9 ) = f r a c 9 23 .


Examples
Imagine you are designing a system where a signal is processed twice through a filter. The function h ( x ) represents the filter, and x is the input signal. Evaluating ( hh ) ( f r a c 7 5 ) tells you the final output signal after processing an initial signal of f r a c 7 5 through the filter twice. This type of function composition is used in signal processing, control systems, and computer science to model cascaded operations.

Answered by GinnyAnswer | 2025-07-03

To evaluate ( hh ) ( 5 7 ​ ) , first calculate h ( 5 7 ​ ) = 9 1 ​ . Then substitute this result into the function again to find h ( 9 1 ​ ) = 23 9 ​ . Thus, ( hh ) ( 5 7 ​ ) = 23 9 ​ .
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Answered by Anonymous | 2025-07-04