https://file.aiquickdraw.com/mathbot/file/17272365114579c9uu5mg.jpg

The problem asks to find the compositions \((h \circ g \circ f)(x)\) and \((f \circ g \circ h)(x)\) given the following functions:

1. \( f(x) = 3x + 1 \)
2. \( g(x) = \frac{1}{x - 1} \)
3. \( h(x) = x^2 - 4 \)

### Finding \((h \circ g \circ f)(x)\):
This represents applying \(f(x)\), then applying \(g(x)\), and finally applying \(h(x)\).

\[
(h \circ g \circ f)(x) = h(g(f(x)))
\]

- First, apply \(f(x)\):
  \[
  f(x) = 3x + 1
  \]
- Now, apply \(g(x)\) to \(f(x)\):
  \[
  g(f(x)) = g(3x + 1) = \frac{1}{(3x + 1) - 1} = \frac{1}{3x}
  \]
- Finally, apply \(h(x)\) to \(g(f(x))\):
  \[
  h(g(f(x))) = h\left(\frac{1}{3x}\right) = \left(\frac{1}{3x}\right)^2 - 4 = \frac{1}{9x^2} - 4
  \]

Thus, 
\[
(h \circ g \circ f)(x) = \frac{1}{9x^2} - 4
\]

### Finding \((f \circ g \circ h)(x)\):
This represents applying \(h(x)\), then applying \(g(x)\), and finally applying \(f(x)\).

\[
(f \circ g \circ h)(x) = f(g(h(x)))
\]

- First, apply \(h(x)\):
  \[
  h(x) = x^2 - 4
  \]
- Now, apply \(g(x)\) to \(h(x)\):
  \[
  g(h(x)) = g(x^2 - 4) = \frac{1}{(x^2 - 4) - 1} = \frac{1}{x^2 - 5}
  \]
- Finally, apply \(f(x)\) to \(g(h(x))\):
  \[
  f(g(h(x))) = f\left(\frac{1}{x^2 - 5}\right) = 3\left(\frac{1}{x^2 - 5}\right) + 1 = \frac{3}{x^2 - 5} + 1
  \]

Thus, 
\[
(f \circ g \circ h)(x) = \frac{3}{x^2 - 5} + 1
\]

### Final Answers:
1. \((h \circ g \circ f)(x) = \frac{1}{9x^2} - 4\)
2. \((f \circ g \circ h)(x) = \frac{3}{x^2 - 5} + 1\)

---

Would you like any more details or further clarifications? Here are 5 related questions for further exploration:

1. How do you evaluate the domain of the function \((h \circ g \circ f)(x)\)?
2. What are the domains of \(g(x)\) and \(h(x)\) separately?
3. How does function composition relate to real-world applications?
4. What happens if we swap the order of compositions, like \((g \circ f \circ h)(x)\)?
5. Can you simplify the expressions further for specific values of \(x\)?

**Tip:** When composing functions, always apply them step-by-step, starting from the innermost function.

Jika f(x) = 3x + 1, g(x) = 1/(x - 1), dan h(x) = x^2 - 4, tentukan (h o g o f)(x) dan (f o g o h)(x)!

Explore detailed steps to solve complex function compositions involving polynomial, rational, and quadratic functions. Understand how to apply (h o g o f)(x) and (f o g o h)(x) with given functions for a thorough grasp of function operations, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation