Math Problem Statement
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
Solution
To find the correct order to stack the machines (functions) so that when the number is dropped into the first machine, the final machine's output is , we need to determine how each function transforms the input and work backward from the desired output.
Functions:
Plan:
We'll proceed by reversing the process, starting with the desired final output of and working backward through each function to see the appropriate sequence.
-
Last machine's output:
The final result is 11. We reverse : Solve for : So the input to the last machine (before ) must be . -
Machine before is
Now, let's find the input to such that the output is .
. We set : Solve for : However, this is not possible, since cannot be negative.
Thus, this suggests that cannot be the second-to-last machine, and we need to try a different order. Let’s try another arrangement.
Working directly by checking
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Math Problem Analysis
Mathematical Concepts
Function Composition
Algebra
Inverse Functions
Formulas
f(x) = √x
g(x) = -(x-2)^2
h(x) = 2^x - 7
k(x) = (-x/2) - 1
Theorems
Inverse Function Theorem
Basic Algebraic Operations
Suitable Grade Level
Grades 10-12
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