Determine the Domain: Analyzing the Function 4x-10 over √(2.5x-10)

Q: Why can't the square root equal zero if it's in the denominator?

Because division by zero is undefined in mathematics! When $ \sqrt{2.5x-10} = 0 $, you're dividing by zero, which breaks the function.

Q: What's the difference between √x ≥ 0 and √x > 0 in denominators?

For denominators, we need strictly greater than zero (>) to avoid division by zero. For numerators or standalone expressions, we can use ≥ 0.

Q: How do I remember when to use > versus ≥?

Memory trick: If the square root is downstairs (denominator), use > to stay away from zero. If it's upstairs or alone, ≥ is usually fine.

Q: What happens at x = 4 exactly?

At x = 4: $ 2.5(4) - 10 = 0 $, so $ \sqrt{0} = 0 $. This makes the denominator zero, creating an undefined expression.

Q: Can I test my domain by plugging in numbers?

Absolutely! Try x = 3 (should fail): $ \sqrt{2.5(3)-10} = \sqrt{-2.5} $ is undefined. Try x = 5 (should work): $ \sqrt{2.5(5)-10} = \sqrt{2.5} $ ✓

Domain Finding with Square Root Denominators

Look at the following function:

$\frac{4x-10}{\sqrt{2.5x-10}}$

What is the domain of the function?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Does the function have a domain? And if so, what is it?

00:03 A root must be for a positive number greater than 0

00:10 Let's isolate X

00:28 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$\frac{4x-10}{\sqrt{2.5x-10}}$

What is the domain of the function?

Step-by-step solution

To find the domain of the function $\frac{4x-10}{\sqrt{2.5x-10}}$ , we need to ensure the expression under the square root is positive since it cannot equal zero or be negative.

Step 1: Set up the inequality based on the square root:

$2.5x - 10 > 0$

Step 2: Solve the inequality for $x$ :

Add 10 to both sides: $2.5x > 10$
Divide both sides by 2.5: $x > \frac{10}{2.5}$
Calculate: $x > 4$

Step 3: Interpret the result:

The domain of the function is all real numbers greater than 4, $x > 4$ , ensuring the expression inside the square root is always positive.

Thus, the correct domain is represented by choice 2: $x > 4$ .

Final Answer

$x > 4$

Key Points to Remember

Essential concepts to master this topic

Rule: Expression under square root in denominator must be positive
Technique: Solve 2.5x - 10 > 0 to get x > 4
Check: Test x = 5: √(2.5(5) - 10) = √2.5 > 0 ✓

Common Mistakes

Avoid these frequent errors

Setting the expression under the square root ≥ 0 instead of > 0
Don't solve 2.5x - 10 ≥ 0 = x ≥ 4! This includes x = 4, which makes the denominator zero and the function undefined. Always use > 0 when the square root is in the denominator to avoid division by zero.

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 the square root equal zero if it's in the denominator?

Because division by zero is undefined in mathematics! When $\sqrt{2.5x-10} = 0$ , you're dividing by zero, which breaks the function.

What's the difference between √x ≥ 0 and √x > 0 in denominators?

For denominators, we need strictly greater than zero (>) to avoid division by zero. For numerators or standalone expressions, we can use ≥ 0.

How do I remember when to use > versus ≥?

Memory trick: If the square root is downstairs (denominator), use > to stay away from zero. If it's upstairs or alone, ≥ is usually fine.

What happens at x = 4 exactly?

At x = 4: $2.5(4) - 10 = 0$ , so $\sqrt{0} = 0$ . This makes the denominator zero, creating an undefined expression.

Can I test my domain by plugging in numbers?

Absolutely! Try x = 3 (should fail): $\sqrt{2.5(3)-10} = \sqrt{-2.5}$ is undefined. Try x = 5 (should work): $\sqrt{2.5(5)-10} = \sqrt{2.5}$ ✓

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