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 66 is dropped into the first machine, the final machine's output is 1111, we need to determine how each function transforms the input and work backward from the desired output.

Functions:

  1. f(x)=xf(x) = \sqrt{x}
  2. g(x)=(x2)2g(x) = -(x - 2)^2
  3. h(x)=2x7h(x) = 2^x - 7
  4. k(x)=x21k(x) = \frac{-x}{2} - 1

Plan:

We'll proceed by reversing the process, starting with the desired final output of 1111 and working backward through each function to see the appropriate sequence.

  1. Last machine's output: k(x)k(x)
    The final result is 11. We reverse k(x)=x21k(x) = \frac{-x}{2} - 1: 11=x2111 = \frac{-x}{2} - 1 Solve for xx: 11+1=x211 + 1 = \frac{-x}{2} 12=x212 = \frac{-x}{2} x=24x = -24 So the input to the last machine (before k(x)k(x)) must be 24-24.

  2. Machine before k(x)k(x) is h(x)h(x)
    Now, let's find the input to h(x)h(x) such that the output is 24-24.
    h(x)=2x7h(x) = 2^x - 7. We set h(x)=24h(x) = -24: 2x7=242^x - 7 = -24 Solve for xx: 2x=24+7=172^x = -24 + 7 = -17 However, this is not possible, since 2x2^x cannot be negative.

Thus, this suggests that h(x)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

Ask a new question for Free

By Image

Drop file here or Click Here to upload

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