Question

What is the domain of this quadratic function?
y=x2+x–6
{x|x ≤ -1/2}
{x|x ≥ -25/4}

Answer 1

To find the domain of the quadratic function \( y = x^2 + x - 6 \), we start by analyzing the expression itself. The function \( y = x^2 + x - 6 \) is a quadratic function, which means it is defined for all real numbers \( x \).

However, the domain may be restricted by the conditions provided. You mentioned two conditions:

1. \( \{ x | x \leq -\frac{1}{2} \} \)
2. \( \{ x | x \geq -\frac{25}{4} \} \)

To determine the overall domain, we examine each condition separately:

1. The first condition limits \( x \) to values less than or equal to \( -\frac{1}{2} \).
2. The second condition allows \( x \) to take values greater than or equal to \( -\frac{25}{4} \) (which is \( -6.25 \)).

Now, we need to identify the overlap or union of these conditions.

The function is defined for all real numbers, but these conditions restrict \( x \) values to certain intervals.

- The first condition states \( x \) can be in the interval \( (-\infty, -\frac{1}{2}] \).
- The second condition states \( x \) can be in the interval \( [-\frac{25}{4}, \infty) \).

Since these intervals do not overlap, the domain of \( y \) takes into account both conditions combined:

Thus, the domain of the function \( y = x^2 + x - 6 \) under the provided conditions is the union of the two intervals:

\[
\text{Domain} = (-\infty, -\frac{1}{2}] \cup [-\frac{25}{4}, \infty)
\]