what are ways to find the domain and range of y = x^2 - 1/ 2

To find the domain and range of the function \( y = \frac{x^2 - 1}{2} \), let's break it down step by step.

### 1. **Finding the Domain**
The domain of a function refers to all the possible values of \( x \) that can be plugged into the function without causing any issues like division by zero or taking the square root of a negative number.

- **Step 1: Check for restrictions on \( x \)**  
  The function \( y = \frac{x^2 - 1}{2} \) is a quadratic function divided by a constant. Since there is no division by \( x \) itself (i.e., there's no \( \frac{1}{x} \) situation) or any square roots, there are no restrictions on \( x \).  
  **Conclusion:** The domain is all real numbers \( \mathbb{R} \).

  \[
  \text{Domain: } (-\infty, \infty)
  \]

### 2. **Finding the Range**
The range of a function refers to all possible values of \( y \) that can be obtained from the function.

- **Step 1: Analyze the function**  
  The function is a quadratic function shifted and scaled. Specifically, \( y = \frac{x^2 - 1}{2} \) is derived from the basic quadratic function \( y = x^2 \), but scaled by \( \frac{1}{2} \) and shifted down by \( \frac{1}{2} \).

- **Step 2: Find the vertex**  
  The vertex of the parabola \( y = x^2 \) is at \( (0, 0) \). After applying the transformation \( y = \frac{x^2 - 1}{2} \), the vertex is shifted to \( (0, -\frac{1}{2}) \).

- **Step 3: Determine the range based on the vertex**  
  Since the parabola opens upwards and the minimum value of the function occurs at the vertex \( (0, -\frac{1}{2}) \), the range of the function starts from \( -\frac{1}{2} \) and extends to infinity.

  \[
  \text{Range: } \left[-\frac{1}{2}, \infty\right)
  \]

### Summary
- **Domain:** \( (-\infty, \infty) \)
- **Range:** \( \left[-\frac{1}{2}, \infty\right) \)

Would you like more details, or do you have any specific questions?

Here are five related questions for further exploration:
1. How does shifting a quadratic function affect its domain and range?
2. What happens to the domain and range if the function is \( y = \frac{x^2 + 1}{2} \)?
3. How can you find the range of a more complex quadratic function?
4. How does changing the coefficient of \( x^2 \) impact the range of the function?
5. Can you find the domain and range of \( y = \frac{1}{x^2} \)?

**Tip:** When determining the range of a quadratic function, always check where the vertex is located and in which direction the parabola opens. This helps in setting the minimum or maximum point of the range.

What are ways to find the domain and range of y = x^2 - 1/2?

Explore the steps to find the domain and range of the quadratic function y = x^2 - 1/2. This guide covers key algebraic concepts, including the properties of quadratic functions, and is suitable for students in Grades 9-11.

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