https://file.aiquickdraw.com/mathbot/file/17287326060375zjxjljr.jpg

Let's solve the given problem. 

We are tasked with expressing \( f(x) \) as a composition of two functions \( g \) and \( h \), such that \( f(x) = (g \circ h)(x) = g(h(x)) \).

Let’s analyze each of the functions provided to express them as compositions:

### (a) \( f(x) = 3\sin(x^2) \)

We can break this down as follows:
- Let \( h(x) = x^2 \) (the inner function).
- Let \( g(x) = 3\sin(x) \) (the outer function).

Thus, \( f(x) = g(h(x)) = 3\sin(x^2) \).

**Answer for part (a)**:
- \( g(x) = 3\sin(x) \)
- \( h(x) = x^2 \)

---

### (b) \( f(x) = 3\sin^2(x) + 4\sin(x) \)

This can be rewritten using composition:
- Let \( h(x) = \sin(x) \) (the inner function).
- Let \( g(x) = 3x^2 + 4x \) (the outer function, where \( x \) represents \( \sin(x) \)).

Thus, \( f(x) = g(h(x)) = 3h(x)^2 + 4h(x) = 3\sin^2(x) + 4\sin(x) \).

**Answer for part (b)**:
- \( g(x) = 3x^2 + 4x \)
- \( h(x) = \sin(x) \)

---

### (c) \( f(x) = (1 + \sin(x^2))^3 \)

This can be broken down as:
- Let \( h(x) = 1 + \sin(x^2) \) (inner function).
- Let \( g(x) = x^3 \) (outer function).

Thus, \( f(x) = g(h(x)) = (1 + \sin(x^2))^3 \).

**Answer for part (c)**:
- \( g(x) = x^3 \)
- \( h(x) = 1 + \sin(x^2) \)

---

### (d) \( f(x) = \sqrt{1 - \sqrt[3]{x}} \)

Here’s how we can express it:
- Let \( h(x) = \sqrt[3]{x} \) (inner function).
- Let \( g(x) = \sqrt{1 - x} \) (outer function).

Thus, \( f(x) = g(h(x)) = \sqrt{1 - \sqrt[3]{x}} \).

**Answer for part (d)**:
- \( g(x) = \sqrt{1 - x} \)
- \( h(x) = \sqrt[3]{x} \)

---

Would you like further explanation on any part of this? Here are some related questions to expand your understanding:

1. How do you identify inner and outer functions when decomposing a function?
2. What are some common functions that appear frequently in function compositions?
3. How do we verify that a composition of functions equals the original function?
4. What are the applications of function compositions in real-world problems?
5. Can all functions be decomposed into compositions of simpler functions?

**Tip**: When identifying compositions, look for nested operations, where one function's output becomes the input of another!

Express f as a composition of two functions; that is, find g and h such that f = g∘h.

Learn how to express the function f(x) as a composition of two functions g and h. This problem covers key mathematical concepts in function composition, trigonometry, and algebra. Suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation