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What is the range of the original function?

The function given is \( f(x) = x^2 + 2 \) with a domain of \( [0, \infty) \).

### Step 1: Analyze the range of the function
The range of a function is the set of all possible output values. To find the range of \( f(x) \), we observe that:
- For \( x \in [0, \infty) \), the term \( x^2 \) is non-negative, meaning \( x^2 \geq 0 \).
- Thus, \( f(x) = x^2 + 2 \geq 2 \).

Since \( f(x) \) is equal to 2 when \( x = 0 \), and as \( x \) increases, \( f(x) \) becomes larger, the range of \( f(x) \) is:

\[
\text{Range of } f(x) = [2, \infty).
\]

Do you want further details on how to find the inverse, or do you have any questions about this explanation?

Here are 5 related questions:
1. How do you find the inverse of a quadratic function?
2. What is the geometric interpretation of a function's inverse?
3. Why is the domain restricted to \( [0, \infty) \) for this function?
4. Can the range of a function affect the existence of its inverse?
5. How does the range of a function help in determining the domain of its inverse?

**Tip**: To ensure the existence of an inverse for a function, the function must be one-to-one (bijective), which often requires restricting its domain.

Consider the function f(x) = x^2 + 2 for the domain [0, ∞). Find f^(-1)(x), where f^(-1) is the inverse of f. Also state the domain of f^(-1) in interval notation.

Learn how to find the inverse of the function f(x) = x^2 + 2 and determine its domain. This problem covers key algebraic concepts such as inverse functions, the one-to-one property, and domain restrictions. Suitable for students in Grades 11-12.

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