Given the three functions below, which expression equals left parenthesis, r, circle, q, circle, m, right parenthesis, left parenthesis, x, right parenthesis(*r*∘*q*∘*m*)(*x*)?

m, left parenthesis, x, right parenthesis, equals, 2, x, minus, 8, q, left parenthesis, x, right parenthesis, equals, square root of, 4, x, end square root, r, left parenthesis, x, right parenthesis, equals, open vertical bar, x, close vertical bar

*m*(*x*)=2*x*−8*q*(*x*)=4*x*​*r*(*x*)=∣*x*∣

We are tasked with determining \( (r \circ q \circ m)(x) \) using the following functions:

- \( m(x) = 2x - 8 \)
- \( q(x) = \sqrt{4x} \)
- \( r(x) = |x| \)

### Steps to solve:

1. **Apply the function \( m(x) \):**
   - Start by applying \( m(x) = 2x - 8 \).
   - The expression becomes:
     \[
     m(x) = 2x - 8
     \]

2. **Apply the function \( q(x) \):**
   - Next, we apply \( q \) to the result of \( m(x) \), which means we input \( m(x) \) into \( q \).
   - Recall that \( q(x) = \sqrt{4x} \), so we substitute \( m(x) \) into \( q \):
     \[
     q(m(x)) = \sqrt{4(2x - 8)} = \sqrt{8x - 32}
     \]

3. **Apply the function \( r(x) \):**
   - Finally, we apply \( r \) to the result of \( q(m(x)) \), which means we input \( q(m(x)) \) into \( r \).
   - Since \( r(x) = |x| \), we take the absolute value of \( q(m(x)) \):
     \[
     r(q(m(x))) = \left| \sqrt{8x - 32} \right|
     \]

### Final Expression:
The expression for \( (r \circ q \circ m)(x) \) is:
\[
(r \circ q \circ m)(x) = \left| \sqrt{8x - 32} \right|
\]

Let me know if you need further clarifications or details on any steps. Here are five related questions:

1. What is the domain of \( (r \circ q \circ m)(x) \)?
2. How would you find the inverse of \( r(x) \), \( q(x) \), or \( m(x) \)?
3. Can you compute \( (r \circ q \circ m)(2) \)?
4. How do you determine whether \( (r \circ q \circ m)(x) \) is continuous?
5. What happens if \( q(x) \) is replaced with a different function, say \( q(x) = x^2 \)?

**Tip:** Always check the domain restrictions after applying functions like square roots or absolute values to avoid undefined expressions.

Given the three functions below, which expression equals (r ∘ q ∘ m)(x)? m(x) = 2x − 8, q(x) = √(4x), r(x) = |x|

Learn how to solve the function composition problem (r ∘ q ∘ m)(x) involving absolute value and square root. We break down each step to show how m(x) = 2x - 8, q(x) = √(4x), and r(x) = |x| combine to give the final expression |√(8x - 32)|. This is suitable for students in Grades 10-12.

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