Determine the Domain of the Radical Function: √(2.5x²-5)

Q: Why can't we have negative values under the square root?

In the real number system, square roots of negative numbers are undefined. We need $ 2.5x^2 - 5 \geq 0 $ to keep the function in the real numbers.

Q: How do I solve inequalities with x²?

When you get $ x^2 \geq 2 $, take the square root of both sides but remember: this gives you two conditions: $ x \geq \sqrt{2} $ OR $ x \leq -\sqrt{2} $.

Q: Why is the domain split into two parts?

The parabola $ y = 2.5x^2 - 5 $ dips below zero between $ -\sqrt{2} $ and $ \sqrt{2} $. The function is only defined where this parabola is above or on the x-axis.

Q: What does the denominator 5 do to the domain?

The denominator 5 is just a constant that doesn't affect the domain. Since 5 ≠ 0, it doesn't create any restrictions. Only the expression under the square root matters for domain.

Q: How can I check if x = 1.5 is in the domain?

Substitute: $ 2.5(1.5)^2 - 5 = 2.5(2.25) - 5 = 5.625 - 5 = 0.625 > 0 $. Since this is positive, x = 1.5 should be in the domain, but \( 1.5 is false, so it's NOT in the domain!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Does the function have a domain of definition? If so, what is it?

00:03 The domain of definition in the denominator is to prevent division by 0

00:06 Therefore, from the denominator's perspective there is no domain of definition, let's check the numerator

00:10 A root must exist for a positive number

00:18 Let's set it equal to 0 to find the solutions

00:23 Let's isolate X

00:37 When extracting a root there are always 2 solutions, positive and negative

00:44 Let's graph to find the domain of definition

00:53 Between the solutions, any value of X necessarily creates a negative root

00:58 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Look at the following function:

$\frac{\sqrt{2.5x^2-5}}{5}$

What is the domain of the function?

2

Step-by-step solution

To solve this problem, first, we determine the condition under the square root function by solving:

$2.5x^2 - 5 \geq 0$ .

This inequality ensures that the expression inside the square root is non-negative, a requirement for the square root function to be defined over real numbers.

Step 1: Simplify the inequality:

First, add 5 to both sides to isolate the term involving $x$ : $2.5x^2 \geq 5$

Step 2: Solve for $x^2$ by dividing both sides by 2.5: $x^2 \geq \frac{5}{2.5}$
Step 3: Simplify the fraction: $x^2 \geq 2$
Step 4: Solve for $x$ by taking the square root of both sides, considering positive and negative solutions: $x \geq \sqrt{2} \quad \text{or} \quad x \leq -\sqrt{2}$

These conditions define the interval for which the original function is defined, corresponding to the original prompt requirement of a non-negative under-the-root value.

Thus, the domain of the function $\frac{\sqrt{2.5x^2-5}}{5}$ is $x \ge \sqrt{2}$ or $x \le -\sqrt{2}$ .

The correct choice among the provided options is:

$x \ge \sqrt{2}, x \le -\sqrt{2}$

3

Final Answer

$x\ge\sqrt{2},x\le-\sqrt{2}$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Expression under square root must be non-negative
Technique: Solve $2.5x^2 - 5 \geq 0$ to get $x^2 \geq 2$
Check: Test x = 2: $2.5(4) - 5 = 5 \geq 0$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting that square roots of negative values create domain restrictions
Don't just ignore the expression under the radical = undefined function values! When the expression inside becomes negative, the square root isn't defined in real numbers. Always set the expression under the radical ≥ 0 and solve the inequality.

Practice Quiz

Test your knowledge with interactive questions

Given the following function:

$\frac{5-x}{2-x}$

Does the function have a domain? If so, what is it?

FAQ

Everything you need to know about this question

Why can't we have negative values under the square root?

+

In the real number system, square roots of negative numbers are undefined. We need $2.5x^2 - 5 \geq 0$ to keep the function in the real numbers.

How do I solve inequalities with x²?

+

When you get $x^2 \geq 2$ , take the square root of both sides but remember: this gives you two conditions: $x \geq \sqrt{2}$ OR $x \leq -\sqrt{2}$ .

Why is the domain split into two parts?

+

The parabola $y = 2.5x^2 - 5$ dips below zero between $-\sqrt{2}$ and $\sqrt{2}$ . The function is only defined where this parabola is above or on the x-axis.

What does the denominator 5 do to the domain?

+

The denominator 5 is just a constant that doesn't affect the domain. Since 5 ≠ 0, it doesn't create any restrictions. Only the expression under the square root matters for domain.

How can I check if x = 1.5 is in the domain?

+

Substitute: $2.5(1.5)^2 - 5 = 2.5(2.25) - 5 = 5.625 - 5 = 0.625 > 0$ . Since this is positive, x = 1.5 should be in the domain, but $1.5 < \sqrt{2} \approx 1.41$ is false, so it's NOT in the domain!

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Determine the Domain of the Radical Function: √(2.5x²-5)

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