State the interval(s) over which the function f(x)=√1/2x2+14x+24 is continuous.

To find the intervals over which the function \( f(x) = \sqrt{\frac{1}{2}x^2 + 14x + 24} \) is continuous, we need to determine where the expression inside the square root is non-negative, since the square root function is only defined for non-negative values.

### Step 1: Set the radicand (the expression inside the square root) greater than or equal to 0:
\[
\frac{1}{2}x^2 + 14x + 24 \geq 0
\]

### Step 2: Solve the quadratic inequality:
Multiply through by 2 to eliminate the fraction:
\[
x^2 + 28x + 48 \geq 0
\]

Next, solve the related quadratic equation:
\[
x^2 + 28x + 48 = 0
\]
Using the quadratic formula:
\[
x = \frac{-28 \pm \sqrt{28^2 - 4(1)(48)}}{2(1)}
\]
\[
x = \frac{-28 \pm \sqrt{784 - 192}}{2}
\]
\[
x = \frac{-28 \pm \sqrt{592}}{2}
\]
Since \( \sqrt{592} \) is not a perfect square, we simplify it:
\[
x = \frac{-28 \pm \sqrt{592}}{2}
\]
This gives us two approximate solutions:
\[
x_1 \approx -25.29, \quad x_2 \approx -2.71
\]

### Step 3: Analyze the intervals:
The solutions divide the number line into three intervals: \( (-\infty, -25.29] \), \( [-25.29, -2.71] \), and \( [-2.71, \infty) \).

To determine where the quadratic expression is non-negative, test points from each interval in the inequality:
- For \( x \in (-\infty, -25.29] \), the quadratic is positive.
- For \( x \in [-25.29, -2.71] \), the quadratic is negative.
- For \( x \in [-2.71, \infty) \), the quadratic is positive.

Thus, the function is continuous on the intervals:
\[
(-\infty, -25.29] \cup [-2.71, \infty)
\]

The function is **continuous** on these intervals because the expression inside the square root is non-negative on them.

---

Would you like further details on the quadratic formula or continuity properties? Here are 5 related questions to explore further:

1. What are the general conditions for continuity of square root functions?
2. How do you find the roots of any quadratic equation using the quadratic formula?
3. How do quadratic inequalities affect the domain of functions?
4. What is the significance of non-negative values for square root functions?
5. Can functions with square roots ever be discontinuous? Why?

**Tip:** Always check the domain of a function involving square roots by ensuring the radicand is non-negative!

State the interval(s) over which the function f(x)=√1/2x2+14x+24 is continuous.

Learn how to determine the intervals where the function f(x) = √(1/2x² + 14x + 24) is continuous by solving a quadratic inequality. This guide covers algebraic concepts, including quadratic formula applications and the conditions for function continuity. Suitable for Grades 10-12 students.

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