Find the Domain of 5/√(x-5): Analyzing Function Restrictions

Q: Why can't x equal 5 if the square root of 0 is 0?

Great question! While $ \sqrt{0} = 0 $ is true, we have division by zero here: $ \frac{5}{\sqrt{0}} = \frac{5}{0} $, which is undefined in mathematics.

Q: What's the difference between ≥ and > in domain problems?

Use > (greater than) when the expression is in a denominator because zero makes it undefined. Use ≥ (greater than or equal) when the square root isn't dividing, like $ \sqrt{x-5} $ by itself.

Q: What if I have multiple restrictions in one function?

Find all the restrictions separately, then combine them using "and". The domain is where all restrictions are satisfied at the same time.

Q: Why do we need the expression under the square root to be positive?

In real numbers, we can't take the square root of negative numbers. Plus, since this square root is in the denominator, it also can't equal zero (division by zero is undefined).

Function Domains with Square Root Denominators

Look at the following function:

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

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

00:09 Let's isolate X

00:14 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:

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

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{5}{\sqrt{x-5}}$ , we must ensure that the expression inside the square root, $x-5$ , is positive. Furthermore, because the square root is in the denominator, $x-5$ must be greater than zero:

Step 1: Set the argument of the square root greater than zero: $x-5 > 0$ .
Step 2: Solve the inequality: Add 5 to both sides to get $x > 5$ .

Since the inequality $x > 5$ ensures that the denominator is neither zero nor negative, it defines the domain of the function. Thus, the function $\frac{5}{\sqrt{x-5}}$ is defined for all real numbers $x$ where $x > 5$ .

Therefore, the domain of the function is $x > 5$ .

Final Answer

$x > 5$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Square root expressions must be positive, never negative
Technique: Set x-5 > 0, then solve: x > 5
Check: Test x = 6: √(6-5) = √1 = 1, so 5/1 = 5 ✓

Common Mistakes

Avoid these frequent errors

Using x ≥ 5 instead of x > 5
Don't include x = 5 in the domain = division by zero! When x = 5, we get 5/√(5-5) = 5/√0 = 5/0, which is undefined. Always exclude values that make the denominator zero.

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 5 if the square root of 0 is 0?

Great question! While $\sqrt{0} = 0$ is true, we have division by zero here: $\frac{5}{\sqrt{0}} = \frac{5}{0}$ , which is undefined in mathematics.

What's the difference between ≥ and > in domain problems?

Use > (greater than) when the expression is in a denominator because zero makes it undefined. Use ≥ (greater than or equal) when the square root isn't dividing, like $\sqrt{x-5}$ by itself.

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

Substitute the number into the expression under the square root. If you get a positive result, it's in the domain. If you get zero or negative, it's not in the domain.

What if I have multiple restrictions in one function?

Find all the restrictions separately, then combine them using "and". The domain is where all restrictions are satisfied at the same time.

Why do we need the expression under the square root to be positive?

In real numbers, we can't take the square root of negative numbers. Plus, since this square root is in the denominator, it also can't equal zero (division by zero is undefined).

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