To solve the problem \((f \circ g) \circ h = (f \circ g \circ h)(x)\), let's break down the operations using the definitions of function composition. 

### Step-by-step Explanation

1. **Understand the Notation**:
   - \(f \circ g\) denotes the composition of functions \(f\) and \(g\), which means \(f(g(x))\).
   - Similarly, \((f \circ g) \circ h\) denotes the composition of the function \(f \circ g\) with \(h\).

2. **Composition of Functions**:
   - \((f \circ g) \circ h\) means applying \(h\) first, then \(g\), and finally \(f\).
   - In terms of function notation, \((f \circ g) \circ h (x) = (f \circ g)(h(x))\).

3. **Break Down the Composition**:
   - Start with the inner function and apply each function step-by-step:
   \[
   (f \circ g) \circ h (x) = f(g(h(x)))
   \]
   This means we:
   - First, apply \(h\) to \(x\): \(h(x)\).
   - Then apply \(g\) to the result: \(g(h(x))\).
   - Finally, apply \(f\) to this result: \(f(g(h(x)))\).

4. **Compare with \((f \circ g \circ h)(x)\)**:
   - By definition, \(f \circ g \circ h\) means apply \(h\), then \(g\), and finally \(f\), which is:
   \[
   (f \circ g \circ h)(x) = f(g(h(x))).
   \]

5. **Conclusion**:
   - Both expressions, \((f \circ g) \circ h(x)\) and \((f \circ g \circ h)(x)\), result in \(f(g(h(x)))\).

Thus, \((f \circ g) \circ h = f \circ g \circ h\) when applied to any input \(x\).

### Summary

The compositions \((f \circ g) \circ h\) and \(f \circ g \circ h\) are equivalent in this context, meaning their outputs are the same for all \(x\).

Would you like more details on function composition or have any questions?

Here are 5 related questions for further exploration:

1. What is the composition of three different functions \(f(x) = x + 1\), \(g(x) = 2x\), and \(h(x) = x^2\)?
2. How do you find the inverse of a composed function like \(f(g(x))\)?
3. Can function composition be associative and commutative? Why or why not?
4. What happens if one of the functions in the composition is not defined for certain values of \(x\)?
5. How do you differentiate a composition of functions, such as \(f(g(h(x)))\)?

**Tip:** Remember, function composition follows the order from right to left when applying functions, so always start with the innermost function!

Explore the concept of function composition with a detailed explanation of (f ∘ g) ∘ h = (f ∘ g ∘ h)(x). This problem breaks down the operations using definitions of function composition and step-by-step examples. Suitable for advanced high school students and anyone interested in advanced mathematics concepts.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation