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The image contains three mathematical questions related to functions. Let's address each one:

1. **Question 1:**
   - **Function:** \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = x^2 \)
   - **Question:** What is the domain of this function?
   - **Answer:** The domain of this function is **all real numbers** \( \mathbb{R} \) because the function \( x^2 \) is defined for any real number \( x \). The correct choice is **\( \mathbb{R} \)**.

2. **Question 2:**
   - **Function:** \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = x^2 \)
   - **Question:** What is the codomain of this function?
   - **Answer:** The codomain is the set the function maps to, which is given as \( \mathbb{R} \). The correct choice is **\( \mathbb{R} \)**.

3. **Question 3:**
   - **Function:** \( f : \mathbb{R} \to \mathbb{R} \) given by \( f(x) = x^2 \)
   - **Question:** What is the range of this function?
   - **Answer:** The range of \( f(x) = x^2 \) includes all non-negative real numbers, since squaring any real number results in a value \( \geq 0 \). Therefore, the range is \( \mathbb{R}_{\geq 0} \), often denoted as \( [0, \infty) \). The correct choice is **\( \mathbb{R}_{\geq 0} \)** or equivalent notation.

Would you like further details on these solutions?

Here are 5 related questions to expand understanding:
1. What is the difference between codomain and range of a function?
2. How would the domain change if the function were \( f(x) = \frac{1}{x} \)?
3. If the function were \( f(x) = x^3 \), what would be the range?
4. Can a function’s range ever exceed its codomain?
5. How does the graph of \( f(x) = x^2 \) help visualize the domain and range?

**Tip:** The domain represents the set of all possible input values, while the range represents the actual output values that the function can produce from the given domain.

Let f : ℝ → ℝ be given by f(x) = x^2. What are the domain, codomain, and range of this function?

This page explores the domain, codomain, and range of the quadratic function f(x) = x^2. Learn how the function maps real numbers and understand key properties such as the non-negative range of the square function.

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