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.