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.