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Let's analyze the parametric function \( g(x) \), which is given as:

\[
g(x) = \begin{cases} 
x = \sqrt{5 + t} \\
y = \sqrt{t - 4}
\end{cases}
\]

We are asked to find the inverse function, and for that, we need to address the following:

i. **Rule of correspondence (inverse function)**  
ii. **Domain of the inverse function**  
iii. **Range (image) of the inverse function**

### Step 1: Solving for \( t \)
We need to find the inverse of the parametric function. First, express \( t \) in terms of \( x \) and \( y \).

1. From \( x = \sqrt{5 + t} \), square both sides to solve for \( t \):
   \[
   x^2 = 5 + t \quad \Rightarrow \quad t = x^2 - 5
   \]

2. From \( y = \sqrt{t - 4} \), square both sides to express \( t \) in terms of \( y \):
   \[
   y^2 = t - 4 \quad \Rightarrow \quad t = y^2 + 4
   \]

Since both expressions for \( t \) are equal, set them equal to each other:
\[
x^2 - 5 = y^2 + 4
\]
\[
x^2 - y^2 = 9
\]

### Step 2: Solving for \( y \)
Now, solve for \( y \) (which represents the inverse function):
\[
y^2 = x^2 - 9 \quad \Rightarrow \quad y = \pm \sqrt{x^2 - 9}
\]

Thus, the inverse function is:
\[
y = \pm \sqrt{x^2 - 9}
\]

### Step 3: Domain of the Inverse
For the square root function \( y = \sqrt{x^2 - 9} \) to be real, the expression inside the square root must be non-negative:
\[
x^2 - 9 \geq 0 \quad \Rightarrow \quad x^2 \geq 9 \quad \Rightarrow \quad x \geq 3 \quad \text{or} \quad x \leq -3
\]
Therefore, the domain of the inverse function is \( (-\infty, -3] \cup [3, \infty) \).

### Step 4: Range of the Inverse
The range of the inverse function corresponds to the values \( y \) can take. Since \( y = \pm \sqrt{x^2 - 9} \), the range is all real numbers \( y \geq 0 \).

---

### Summary:
- i. **Inverse function**: \( g^{-1}(x) = \pm \sqrt{x^2 - 9} \)
- ii. **Domain of the inverse**: \( (-\infty, -3] \cup [3, \infty) \)
- iii. **Range of the inverse**: \( [0, \infty) \)

Would you like more details on any part of the solution? Here are some related questions:

1. How can we verify if a function has an inverse?
2. Why do we restrict the domain when finding the inverse of square root functions?
3. How can we graph the inverse function?
4. How does the behavior of a parametric function change when reversed?
5. What is the relationship between the domain and range of a function and its inverse?

**Tip:** Always check the domain restrictions carefully when dealing with square roots and inverses!

Obtain for the inverse function of g(x) expressed in parametric form: i. Rule of correspondence ii. Domain iii. Range

Learn how to find the inverse of a parametric function expressed in terms of t, including its rule of correspondence, domain, and range. This detailed solution covers key algebraic concepts and is suitable for college-level calculus students.

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