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.