In what order should you stack the machines so that when 6 is dropped into the first machine, and all four machines have had their effect, the last machine's output is 11? -f(x)= √x  -g(x)= -(x-2)^2      -h(x)= 2^x - 7  -k(x)=(-x/2)-1

To find the correct order to stack the machines (functions) so that when the number \(6\) is dropped into the first machine, the final machine's output is \(11\), we need to determine how each function transforms the input and work backward from the desired output.

### Functions:
1. \( f(x) = \sqrt{x} \)
2. \( g(x) = -(x - 2)^2 \)
3. \( h(x) = 2^x - 7 \)
4. \( k(x) = \frac{-x}{2} - 1 \)

### Plan:
We'll proceed by reversing the process, starting with the desired final output of \(11\) and working backward through each function to see the appropriate sequence.

1. **Last machine's output: \( k(x) \)**  
   The final result is 11. We reverse \( k(x) = \frac{-x}{2} - 1 \):
   \[
   11 = \frac{-x}{2} - 1
   \]
   Solve for \( x \):
   \[
   11 + 1 = \frac{-x}{2}
   \]
   \[
   12 = \frac{-x}{2}
   \]
   \[
   x = -24
   \]
   So the input to the last machine (before \( k(x) \)) must be \( -24 \).

2. **Machine before \( k(x) \) is \( h(x) \)**  
   Now, let's find the input to \( h(x) \) such that the output is \( -24 \).  
   \( h(x) = 2^x - 7 \). We set \( h(x) = -24 \):
   \[
   2^x - 7 = -24
   \]
   Solve for \( x \):
   \[
   2^x = -24 + 7 = -17
   \]
   However, this is not possible, since \( 2^x \) cannot be negative. 

Thus, this suggests that \( h(x) \) cannot be the second-to-last machine, and we need to try a different order. Let’s try another arrangement.

### Working directly by checking

In what order should you stack the machines so that when 6 is dropped into the first machine, and all four machines have had their effect, the last machine's output is 11? -f(x)= √x -g(x)= -(x-2)^2 -h(x)= 2^x - 7 -k(x)=(-x/2)-1

Solve a challenging math problem using function composition. Learn how to stack functions f(x)=√x, g(x)=-(x-2)^2, h(x)=2^x-7, and k(x)=(-x/2)-1 in the correct order to achieve an output of 11. Suitable for high school students in Grades 10-12.

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