https://file.aiquickdraw.com/mathbot/file/1727186293421p18icgg3.jpg

Let's break down the tasks step by step based on the given image.

### 1. Functions and their Domains:
- \( f(x) = 9x^2 + 6 \), where the domain \( M_f = [-\infty, -1] \).
- \( g(x) = -\frac{1}{3} \sqrt{x - 6} \), where the domain \( M_g = [7, \infty[ \).

The task is to find the composition of these functions, along with the associated domains, and also the inverse of another function \( h(x) \).

### 2. Composition of Functions:
You are asked to compute \( f \circ g \) and \( g \circ f \), then find the domains \( M_{f \circ g} \) and \( M_{g \circ f} \).

#### **a. Finding \( f \circ g \) and its domain**:
- \( f \circ g(x) = f(g(x)) \), which means you substitute \( g(x) \) into \( f(x) \).
- Given that \( g(x) = -\frac{1}{3} \sqrt{x - 6} \), substitute this into \( f(x) = 9x^2 + 6 \).
- Find the domain where this composition is valid. It will be the intersection of the domains of \( f \) and \( g \), considering the range of \( g(x) \) must fit within the domain of \( f(x) \).

#### **b. Finding \( g \circ f \) and its domain**:
- \( g \circ f(x) = g(f(x)) \), which means you substitute \( f(x) \) into \( g(x) \).
- Substitute \( f(x) = 9x^2 + 6 \) into \( g(x) = -\frac{1}{3} \sqrt{x - 6} \).
- Similarly, find the domain \( M_{g \circ f} \), ensuring the input of \( f(x) \) leads to valid outputs in the domain of \( g(x) \).

### 3. Inverse of \( h(x) \):
The function \( h(x) = \sqrt{x^2 + 28} - 5 \), with \( x \in [0, 2] \), is given. You are asked to find the inverse function \( h^{-1}(x) \) and its domain \( M_{h^{-1}} \).

- To find the inverse, solve \( y = \sqrt{x^2 + 28} - 5 \) for \( x \) in terms of \( y \).
- Determine the domain where the inverse function is valid, ensuring you respect the domain restriction of \( h(x) \).

---

I will now work through these calculations and provide the solutions.Here are the results from the calculations:

### 1. Composition \( f \circ g(x) \):
- \( f \circ g(x) = 9\left(-\frac{1}{3} \sqrt{x - 6}\right)^2 + 6 = x \).
- The domain \( M_{f \circ g} \) is where \( g(x) \) is valid within the domain of \( f(x) \), which would be \( x \in [7, \infty[ \).

### 2. Composition \( g \circ f(x) \):
- \( g \circ f(x) = -\frac{1}{3} \sqrt{9x^2 + 6 - 6} = -\sqrt{x^2} \).
- This simplifies to \( -|x| \), with the domain \( M_{g \circ f} \) requiring \( 9x^2 + 6 \geq 6 \), so \( x \in [-1, -\infty] \).

### 3. Inverse of \( h(x) \):
- The inverse of \( h(x) = \sqrt{x^2 + 28} - 5 \) is:
  \[
  h^{-1}(x) = \pm \sqrt{(x + 5)^2 - 28}
  \]
- Given the domain of \( h(x) \) is \( [0, 2] \), we choose the positive branch \( h^{-1}(x) = \sqrt{(x + 5)^2 - 28} \).
- The domain of the inverse, \( M_{h^{-1}} \), depends on the range of \( h(x) \), which is the output of \( h(x) \). This leads to \( x \in [-5, 0] \).

---

Would you like further details on any step? Here are 5 related questions:

1. How do we determine the range of the composite function \( f \circ g \)?
2. How does the absolute value affect the result of \( g \circ f \)?
3. Why do we select the positive root for the inverse of \( h(x) \)?
4. Can we visualize these function compositions graphically?
5. How can we verify the domain restrictions for these compositions?

**Tip**: When working with composite functions, always ensure the output of one function is valid within the domain of the next function in the composition.

Find the composite functions f ◦ g, g ◦ f and their domains. Also, find the inverse of the function h(x) = √(x² + 28) - 5 and its domain.

This problem involves finding the composite functions f ◦ g, g ◦ f, and their domains. Additionally, the inverse of the function h(x) = √(x² + 28) - 5 is found, along with its domain. Learn how to compute and understand function compositions and inverses, with a focus on domain restrictions.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation