Determine the Domain: Analyzing the Function 23/√x

Q: Why can't x equal zero if square roots allow zero?

Good catch! While $ \sqrt{0} = 0 $ is perfectly valid, having zero in the denominator of a fraction makes the entire expression undefined. Division by zero is never allowed in mathematics.

Q: How do I write the domain correctly?

The domain is x > 0, which means all positive real numbers. You can also write this as $ (0, \infty) $ in interval notation or "all real numbers greater than zero."

Q: What if I had √(x-5) in the denominator instead?

Then you'd need x - 5 > 0, which means x > 5. The expression under the square root must be positive (not just non-negative) to avoid division by zero.

Domain Restrictions with Square Root Denominators

Look at the following function:

$\frac{23}{\sqrt{x}}$

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

00:06 Does the function have a domain? If it does, what is it?

00:10 To find the domain, remember, division by zero is not allowed.

00:15 Since the unknown is under a root, it must be positive.

00:20 And that solves our question! Great job!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$\frac{23}{\sqrt{x}}$

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{23}{\sqrt{x}}$ , we must ensure the function is defined for all values in its domain. The expression involves a square root and a division.

First, consider the square root, $\sqrt{x}$ . This is only defined for $x \geq 0$ . Therefore, initially, $x$ must be non-negative.
Second, because the square root is in the denominator of a fraction, $\sqrt{x}$ must not equal zero to avoid division by zero. Thus, $x$ must be strictly greater than 0.

Combining these conditions, we find that the domain of the function is $x > 0$ .

Therefore, the domain of the function is $x > 0$ , which corresponds to choice 3 from the provided options.

Final Answer

$x > 0$

Key Points to Remember

Essential concepts to master this topic

Square Root Rule: Expression under square root must be non-negative
Division Rule: Denominator cannot equal zero, so $\sqrt{x} \neq 0$
Check: Test x = 1: $\frac{23}{\sqrt{1}} = \frac{23}{1} = 23$ works ✓

Common Mistakes

Avoid these frequent errors

Including x = 0 in the domain
Don't write x ≥ 0 as the domain = division by zero error! When x = 0, √0 = 0 makes the denominator zero, which is undefined. Always exclude values that make denominators zero, so the domain is x > 0.

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 zero if square roots allow zero?

Good catch! While $\sqrt{0} = 0$ is perfectly valid, having zero in the denominator of a fraction makes the entire expression undefined. Division by zero is never allowed in mathematics.

What about negative numbers under the square root?

In real numbers, we cannot take the square root of negative values. So x must be non-negative for $\sqrt{x}$ to be defined, but we also need x > 0 to avoid division by zero.

How do I write the domain correctly?

The domain is x > 0, which means all positive real numbers. You can also write this as $(0, \infty)$ in interval notation or "all real numbers greater than zero."

Does the number 23 in the numerator affect the domain?

No! The numerator 23 is just a constant and doesn't create any restrictions. Only the denominator $\sqrt{x}$ determines what values of x are allowed.

What if I had √(x-5) in the denominator instead?

Then you'd need x - 5 > 0, which means x > 5. The expression under the square root must be positive (not just non-negative) to avoid division by zero.

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