Analyzing the Domain of 3/√(x-10): Understanding Function Constraints

Q: Why can't x equal 10 if the square root of 0 exists?

While $ \sqrt{0} = 0 $ is mathematically valid, having zero in the denominator makes the entire fraction undefined. Remember: you can never divide by zero!

Q: How is this different from just √(x-10) without the fraction?

For $ \sqrt{x-10} $ alone, the domain would be $ x \geq 10 $. But with the fraction $ \frac{3}{\sqrt{x-10}} $, we must exclude x = 10 to avoid division by zero.

Q: What does the domain x > 10 look like on a number line?

Draw an open circle at x = 10 (meaning 10 is not included) and shade everything to the right. This shows all numbers greater than 10 are allowed.

Q: Can I write the domain as (10, ∞) instead?

Yes! The interval notation (10, ∞) means exactly the same as x > 10. The parenthesis shows 10 is not included, and ∞ means it continues forever.

Q: What happens if I try to plug in x = 5?

If x = 5, then x - 10 = -5. Since $ \sqrt{-5} $ is not a real number, x = 5 is not in the domain. This confirms we need x > 10.

Function Domains with Square Root Denominators

Look at the following function:

$\frac{3}{\sqrt{x-10}}$

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 The root must be for a positive number greater than 0

00:10 Let's isolate X

00:15 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:

$\frac{3}{\sqrt{x-10}}$

What is the domain of the function?

Step-by-step solution

To find the domain of the function $f(x) = \frac{3}{\sqrt{x-10}}$ , follow these steps:

First, ensure the expression under the square root, $x-10$ , is non-negative. This gives $x-10 \geq 0$ , or equivalently, $x \geq 10$ .
Second, the denominator of the function, $\sqrt{x-10}$ , cannot be zero. This means $\sqrt{x-10} \neq 0$ , leading to $x-10 \neq 0$ . Therefore, $x \neq 10$ .

Combining these conditions, the value of $x$ must satisfy $x > 10$ . This ensures both the definition of the square root and the non-zero nature of the denominator.

The correct domain of the function is $x > 10$ .

Final Answer

$x > 10$

Key Points to Remember

Essential concepts to master this topic

Rule: Square root expressions require non-negative inputs only
Technique: Set x - 10 > 0 (not ≥ 0) to avoid zero denominator
Check: Test x = 11: √(11-10) = 1, function equals 3/1 = 3 ✓

Common Mistakes

Avoid these frequent errors

Including x = 10 in the domain
Don't write x ≥ 10 when there's a square root in the denominator = division by zero! When x = 10, √(x-10) = √0 = 0, making 3/0 undefined. Always use x > 10 to exclude the boundary value.

Practice Quiz

Test your knowledge with interactive questions

$22(\frac{2}{x}-1)=30$

What is the domain of the equation above?

FAQ

Everything you need to know about this question

Why can't x equal 10 if the square root of 0 exists?

While $\sqrt{0} = 0$ is mathematically valid, having zero in the denominator makes the entire fraction undefined. Remember: you can never divide by zero!

How is this different from just √(x-10) without the fraction?

For $\sqrt{x-10}$ alone, the domain would be $x \geq 10$ . But with the fraction $\frac{3}{\sqrt{x-10}}$ , we must exclude x = 10 to avoid division by zero.

What does the domain x > 10 look like on a number line?

Draw an open circle at x = 10 (meaning 10 is not included) and shade everything to the right. This shows all numbers greater than 10 are allowed.

Can I write the domain as (10, ∞) instead?

Yes! The interval notation (10, ∞) means exactly the same as x > 10. The parenthesis shows 10 is not included, and ∞ means it continues forever.

What happens if I try to plug in x = 5?

If x = 5, then x - 10 = -5. Since $\sqrt{-5}$ is not a real number, x = 5 is not in the domain. This confirms we need x > 10.

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