Determining the Domain: Rational Function with a Square Root Denominator

Q: Why can't x equal 4 if the square root of 0 exists?

While $ \sqrt{0} = 0 $ is defined, having zero in the denominator makes the entire fraction undefined. Division by zero is never allowed in mathematics!

Q: What's the difference between domain restrictions for square roots and denominators?

For square roots alone, we need non-negative values (≥ 0). But for square roots in denominators, we need positive values (> 0) to avoid division by zero.

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

Ask yourself: "Will this value make any part undefined?" If yes, exclude it with strict inequality (>). If the expression stays defined, include it with ≥.

Q: Can I plug in x = 4 to check my answer?

No! Since x = 4 is not in the domain, plugging it in would give division by zero. Always test values that are actually inside your domain, like x = 5.

Q: What if I have multiple restrictions on the same function?

Find all restrictions separately, then take their intersection. The domain must satisfy every single restriction simultaneously.

Domain Restrictions with Square Root Denominators

Look at the following function:

$\frac{2x+2}{\sqrt{2x-8}}$

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

00:09 Does this function have a domain? If yes, what is it? Let's explore.

00:14 Remember, a root must be a positive number greater than zero.

00:20 Next, let's isolate X. We can do this step by step.

00:37 Great job! And that's the solution to our question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$\frac{2x+2}{\sqrt{2x-8}}$

What is the domain of the function?

Step-by-step solution

To solve this problem, we need to determine where the function $f(x) = \frac{2x+2}{\sqrt{2x-8}}$ is defined.

For the fraction to be defined, the denominator cannot be zero, and for the square root to be defined, the radicand (the expression inside the square root) must be non-negative.

Therefore, we need to solve the inequality:

$2x - 8 \geq 0$

Solving this inequality involves the following steps:

Add 8 to both sides: $2x \geq 8$
Divide both sides by 2: $x \geq 4$

However, if $x = 4$ , the expression inside the square root is zero, making the denominator zero and the overall expression undefined.

As a result, the domain of the function is $x > 4$ .

Therefore, the domain of the function is $x > 4$ .

Final Answer

$x > 4$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Expression under square root must be positive for denominators
Technique: Solve $2x - 8 > 0$ to get $x > 4$
Check: Test $x = 5$ : $\sqrt{2(5)-8} = \sqrt{2}$ works ✓

Common Mistakes

Avoid these frequent errors

Using ≥ instead of > for square root denominators
Don't write $x ≥ 4$ = division by zero! When x = 4, the denominator $\sqrt{2(4)-8} = \sqrt{0} = 0$ makes the function undefined. Always use strict inequality > to exclude values that make denominators zero.

Practice Quiz

Test your knowledge with interactive questions

$\frac{6}{x+5}=1$

What is the field of application of the equation?

FAQ

Everything you need to know about this question

Why can't x equal 4 if the square root of 0 exists?

While $\sqrt{0} = 0$ is defined, having zero in the denominator makes the entire fraction undefined. Division by zero is never allowed in mathematics!

What's the difference between domain restrictions for square roots and denominators?

For square roots alone, we need non-negative values (≥ 0). But for square roots in denominators, we need positive values (> 0) to avoid division by zero.

How do I remember when to use > versus ≥?

Ask yourself: "Will this value make any part undefined?" If yes, exclude it with strict inequality (>). If the expression stays defined, include it with ≥.

Can I plug in x = 4 to check my answer?

No! Since x = 4 is not in the domain, plugging it in would give division by zero. Always test values that are actually inside your domain, like x = 5.

What if I have multiple restrictions on the same function?

Find all restrictions separately, then take their intersection. The domain must satisfy every single restriction simultaneously.

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