Evaluate $(h h)\left(\frac{1}{2}\right)$ given that $h(x)=\frac{1}{2 x+4}$ Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. (hh) $\left(\frac{1}{2}\right)=$ $\square$ (Simplify your answer.) B. The answer is undefined.

First, find h ( f r a c 1 2 ) by substituting x = f r a c 1 2 into h ( x ) = f r a c 1 2 x + 4 , which gives h ( f r a c 1 2 ) = f r a c 1 5 .
Then, find h ( h ( f r a c 1 2 )) = h ( f r a c 1 5 ) by substituting x = f r a c 1 5 into h ( x ) , which gives h ( f r a c 1 5 ) = f r a c 1 2 ( f r a c 1 5 ) + 4 = f r a c 5 22 .
Therefore, ( hh ) ( f r a c 1 2 ) = f r a c 5 22 .
The final answer is f r a c 5 22 ​ .

Explanation

Understanding the Problem We are given the function h ( x ) = f r a c 1 2 x + 4 and we need to evaluate ( hh ) ( f r a c 1 2 ) , which means we need to find h ( h ( f r a c 1 2 )) . In other words, we need to evaluate the composite function h ( h ( x )) at x = f r a c 1 2 .

Evaluating h(1/2) First, we evaluate h ( f r a c 1 2 ) by substituting x = f r a c 1 2 into the expression for h ( x ) : h ( f r a c 1 2 ) = f r a c 1 2 ( f r a c 1 2 ) + 4 = f r a c 1 1 + 4 = f r a c 1 5 So, h ( f r a c 1 2 ) = f r a c 1 5 .

Evaluating h(h(1/2)) Next, we evaluate h ( h ( f r a c 1 2 )) = h ( f r a c 1 5 ) by substituting x = f r a c 1 5 into the expression for h ( x ) : h ( f r a c 1 5 ) = f r a c 1 2 ( f r a c 1 5 ) + 4 = f r a c 1 f r a c 2 5 + 4 = f r a c 1 f r a c 2 + 20 5 = f r a c 1 f r a c 22 5 = f r a c 5 22 So, h ( h ( f r a c 1 2 )) = f r a c 5 22 .

Final Answer Therefore, ( hh ) ( f r a c 1 2 ) = f r a c 5 22 .


Examples
Composite functions are used in many real-world applications. For example, in manufacturing, a company might have a function that converts raw materials into a semi-finished product, and another function that converts the semi-finished product into a finished product. The composite of these two functions would describe the entire manufacturing process from raw materials to finished product. Similarly, in computer graphics, transformations like scaling, rotation, and translation can be represented as functions, and applying multiple transformations in sequence is equivalent to composing those functions.

Answered by GinnyAnswer | 2025-07-03

To find (h h)igg(\frac{1}{2}igg) we first evaluate higg(\frac{1}{2}igg) which gives us 5 1 ​ . Next, we evaluate higg(\frac{1}{5}igg) , leading us to the final result of 22 5 ​ . Thus, the answer is 22 5 ​ .
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Answered by Anonymous | 2025-07-04