Find the Domain of 3/√(2x-3): Square Root Function Analysis

Q: Why can't x equal 1.5 if the square root exists?

Even though $ \sqrt{0} = 0 $ exists, we have division by zero in the denominator! The function $ \frac{3}{\sqrt{2x-3}} $ becomes $ \frac{3}{0} $ when x = 1.5, which is undefined.

Q: What's the difference between domain and range?

Domain is all possible x-values (input), while range is all possible y-values (output). For this problem, we're finding which x-values make the function work.

Q: How do I write the domain in interval notation?

The domain $ x > 1.5 $ is written as (1.5, ∞) in interval notation. Use parentheses ( ) because 1.5 is not included in the domain.

Q: What if the square root wasn't in the denominator?

If we had just $ \sqrt{2x-3} $ (not in denominator), then we'd use $ x \geq 1.5 $ because $ \sqrt{0} = 0 $ is allowed when it's not dividing.

Q: Can I use decimals or should I use fractions?

Both are correct! $ x > 1.5 $ and $ x > \frac{3}{2} $ mean exactly the same thing. Use whichever form is easier for you to work with.

Domain Restrictions with Square Root Denominators

Look at the following function:

$\frac{3}{\sqrt{2x-3}}$

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 The root must be for a positive number greater than 0

00:11 Let's isolate X

00:21 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:

$\frac{3}{\sqrt{2x-3}}$

What is the domain of the function?

Step-by-step solution

To solve this problem, we need to determine the domain of the function given by $\frac{3}{\sqrt{2x-3}}$ . The fraction is undefined whenever the denominator is zero, and the square root requires the expression inside to be positive.

Let's break down the steps:

Since the square root is in the denominator, $2x - 3$ must be strictly greater than zero.
Set up the inequality: $2x - 3 > 0$ .
Solve the inequality for $x$ :

Add 3 to both sides: $2x > 3$ .
Divide both sides by 2: $x > 1.5$ .

The domain of the function is all real numbers $x$ such that $x > 1.5$ . Therefore, we write the domain as $x > 1.5$ .

Hence, the correct choice among the given options is: $x > 1.5$ which corresponds to choice number 3.

Final Answer

$x > 1.5$

Key Points to Remember

Essential concepts to master this topic

Rule: Square root denominators require the radicand to be positive
Technique: Solve $2x - 3 > 0$ to get $x > 1.5$
Check: Test $x = 2$ : $2(2) - 3 = 1 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Using ≥ instead of > for the inequality
Don't write x ≥ 1.5 when the square root is in the denominator = division by zero! When x = 1.5, the denominator becomes √0 = 0, making the fraction undefined. Always use strict inequality > when square roots are in denominators.

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 1.5 if the square root exists?

Even though $\sqrt{0} = 0$ exists, we have division by zero in the denominator! The function $\frac{3}{\sqrt{2x-3}}$ becomes $\frac{3}{0}$ when x = 1.5, which is undefined.

What's the difference between domain and range?

Domain is all possible x-values (input), while range is all possible y-values (output). For this problem, we're finding which x-values make the function work.

How do I write the domain in interval notation?

The domain $x > 1.5$ is written as (1.5, ∞) in interval notation. Use parentheses ( ) because 1.5 is not included in the domain.

What if the square root wasn't in the denominator?

If we had just $\sqrt{2x-3}$ (not in denominator), then we'd use $x \geq 1.5$ because $\sqrt{0} = 0$ is allowed when it's not dividing.

Can I use decimals or should I use fractions?

Both are correct! $x > 1.5$ and $x > \frac{3}{2}$ mean exactly the same thing. Use whichever form is easier for you to work with.

How do I check my domain answer?

Pick a test value greater than 1.5 (like x = 2) and substitute: $\frac{3}{\sqrt{2(2)-3}} = \frac{3}{\sqrt{1}} = \frac{3}{1} = 3$ . It works! Try x = 1: $\sqrt{2(1)-3} = \sqrt{-1}$ - not real!

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