Uncover the Domain of a Radical Rational Function: (x+7)/√(x-7)

Q: Why can't x equal 7 if the square root of 0 is defined?

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

Q: What's the difference between x > 7 and x ≥ 7 for this problem?

The symbol $ x \geq 7 $ includes x = 7, but at x = 7 our denominator becomes zero. We need $ x > 7 $ to exclude x = 7 and keep the function defined.

Q: What if the expression was √(x-7) in the numerator instead?

If the square root was in the numerator, we'd only need $ x - 7 \geq 0 $ or $ x \geq 7 $, since zero in the numerator is allowed (it just makes the whole fraction equal zero).

Q: Are there other restrictions I should check for this function?

For $ \frac{x+7}{\sqrt{x-7}} $, the only restriction comes from the square root in the denominator. The numerator $ x + 7 $ has no restrictions since it's just a linear expression.

Domain Restrictions with Square Root Denominators

Look at the following function:

$\frac{x+7}{\sqrt{x-7}}$

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? If so, what is it?

00:04 Root must be for a positive number greater than 0

00:08 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:

$\frac{x+7}{\sqrt{x-7}}$

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{x+7}{\sqrt{x-7}}$ , we need to ensure the denominator remains defined and non-zero. As follows:

First, focus on the denominator $\sqrt{x-7}$ . The expression under the square root, $x-7$ , must be greater than zero for the square root to be defined and not produce zero in the denominator:

$x - 7 > 0$

This simplifies to:

$x > 7$

Since the expression under the square root must always be positive for this rational function to be defined, and $\sqrt{x-7}$ in the denominator implies it cannot equal zero, our analysis is complete. Consequently, the domain of the function is the set of all $x$ such that:

The domain of the function is $x > 7$ .

Final Answer

$x > 7$

Key Points to Remember

Essential concepts to master this topic

Square Root Rule: Expression under square root must be non-negative
Technique: For $x - 7 > 0$ , solve to get $x > 7$
Check: Test x = 8: $\sqrt{8-7} = \sqrt{1} = 1$ (defined) ✓

Common Mistakes

Avoid these frequent errors

Including x = 7 in the domain
Don't write x ≥ 7 because at x = 7, the denominator √(x-7) = √0 = 0, creating division by zero! This makes the function undefined at that point. Always exclude values that make denominators zero, so use x > 7.

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 7 if the square root of 0 is defined?

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

What's the difference between x > 7 and x ≥ 7 for this problem?

The symbol $x \geq 7$ includes x = 7, but at x = 7 our denominator becomes zero. We need $x > 7$ to exclude x = 7 and keep the function defined.

How do I check if a value is in the domain?

Substitute the value into the expression under the square root. If $x - 7 > 0$ , then it's in the domain. For example: x = 10 gives $10 - 7 = 3 > 0$ ✓

What if the expression was √(x-7) in the numerator instead?

If the square root was in the numerator, we'd only need $x - 7 \geq 0$ or $x \geq 7$ , since zero in the numerator is allowed (it just makes the whole fraction equal zero).

Are there other restrictions I should check for this function?

For $\frac{x+7}{\sqrt{x-7}}$ , the only restriction comes from the square root in the denominator. The numerator $x + 7$ has no restrictions since it's just a linear expression.

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