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To solve question 14, we are tasked with finding the formulas for \( g_2(x) \) and \( g_2^{-1}(x) \) and stating the domain and range of each function. Let's work through this:

### Observing the Graph:
We can see that the given graph is a parabola, \( y = x^2 + 1 \), and the functions \( g_1(x) \), \( g_2(x) \), and their inverses are also plotted.

For \( g_2(x) \), it appears to be the **lower branch** of the parabola, and \( g_2^{-1}(x) \) is the reflection of \( g_2(x) \) across the line \( y = x \).

### Step-by-Step Solution for \( g_2(x) \):

#### 1. **Formula for \( g_2(x) \)**:
Since \( g_2(x) \) represents the lower branch of the parabola, it must correspond to:
\[
g_2(x) = -\sqrt{x - 1}
\]
This is because the equation \( y = x^2 + 1 \) represents the parabola, and the lower branch is obtained by taking the negative square root (inverting the sign).

#### 2. **Domain and Range of \( g_2(x) \)**:
- **Domain**: Since \( g_2(x) = -\sqrt{x - 1} \), the expression inside the square root must be non-negative. Therefore, the domain is \( x \geq 1 \) or \( [1, \infty) \).
- **Range**: Because we are taking the negative square root, the values of \( g_2(x) \) will be non-positive. As \( x \to 1 \), \( g_2(x) \to 0 \), and as \( x \to \infty \), \( g_2(x) \to -\infty \). Therefore, the range is \( (-\infty, 0] \).

### Step-by-Step Solution for \( g_2^{-1}(x) \):

#### 1. **Formula for \( g_2^{-1}(x) \)**:
The inverse function \( g_2^{-1}(x) \) is found by solving \( y = -\sqrt{x - 1} \) for \( x \) in terms of \( y \):
\[
y = -\sqrt{x - 1} \implies -y = \sqrt{x - 1} \implies x = y^2 + 1
\]
Thus, \( g_2^{-1}(x) = x^2 + 1 \).

#### 2. **Domain and Range of \( g_2^{-1}(x) \)**:
- **Domain**: The domain of \( g_2^{-1}(x) \) is the range of \( g_2(x) \), which is \( (-\infty, 0] \).
- **Range**: The range of \( g_2^{-1}(x) \) is the domain of \( g_2(x) \), which is \( [1, \infty) \).

### Final Answer:
- \( g_2(x) = -\sqrt{x - 1} \), with domain \( [1, \infty) \) and range \( (-\infty, 0] \).
- \( g_2^{-1}(x) = x^2 + 1 \), with domain \( (-\infty, 0] \) and range \( [1, \infty) \).

Let me know if you want further details or have any questions. 

Here are some related questions:
1. How is the inverse of a function related to its graph?
2. Why does the lower branch of the parabola correspond to \( g_2(x) \)?
3. What is the general process for finding the inverse of a function?
4. How does the domain of a function relate to the range of its inverse?
5. What are the conditions for a function to have an inverse?

**Tip**: Always check if the original function is one-to-one before finding its inverse; otherwise, restrict its domain accordingly.

Find formulas for g_2(x) and g_2^{-1}(x). State the domain and range of each function.

Learn how to find the formula for g_2(x) and its inverse g_2^{-1}(x). This problem explores functions, their inverses, and how to determine the domain and range for each. Key mathematical concepts include algebra, functions, and quadratic equations, suitable for Grades 11-12.

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