Determine the Domain: Analyzing the Function 10x Divided by 1/2

Q: Why isn't x ≠ 1/2 part of the domain restriction?

Great question! The denominator is the constant 1/2, not the variable x. Since 1/2 ≠ 0, there's no division by zero issue. Domain restrictions only occur when the variable could make a denominator zero.

Q: How do I know when to simplify before finding the domain?

Always simplify first! The original form might look complicated, but once you simplify $ \frac{10x}{\frac{1}{2}} $ to 20x, you can clearly see it's just a linear function with no restrictions.

Q: How do I remember the rule for dividing by fractions?

Think "flip and multiply"! Dividing by $ \frac{1}{2} $ is the same as multiplying by $ \frac{2}{1} = 2 $. So $ \frac{10x}{\frac{1}{2}} = 10x \times 2 $.

Q: Can a linear function ever have domain restrictions?

In standard form like f(x) = mx + b, linear functions have all real numbers as their domain. Domain restrictions only appear in rational, radical, or logarithmic functions where certain x-values cause undefined operations.

Domain Analysis with Simplified Linear Functions

Look at the following function:

$\frac{10x}{\frac{1}{2}}$

What is the domain of the function?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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 To find the domain, remember that division by 0 is not allowed

00:07 The denominator is a constant number different from 0

00:10 Therefore there is no domain restriction

00:13 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:

$\frac{10x}{\frac{1}{2}}$

What is the domain of the function?

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Simplify the expression.

Given the function:

$f(x) = \frac{10x}{\frac{1}{2}}$

We can simplify this expression by multiplying by the reciprocal of the denominator:

$f(x) = 10x \times 2 = 20x$

Step 2: Determine the domain.

Since $f(x) = 20x$ is a linear function, it is defined for all real numbers. There are no restrictions on $x$ since no division by zero or any undefined operations are present.

Conclusion: The domain of the function is all real numbers. This corresponds to choice :

All real numbers

Therefore, the domain of the function is all real numbers.

Final Answer

All real numbers

Key Points to Remember

Essential concepts to master this topic

Simplification Rule: Division by fraction equals multiplication by its reciprocal
Technique: $\frac{10x}{\frac{1}{2}} = 10x \times 2 = 20x$
Check: Linear functions like 20x have domain of all real numbers ✓

Common Mistakes

Avoid these frequent errors

Restricting domain based on original denominator
Don't think x ≠ 1/2 because 1/2 appears in denominator = wrong domain! The constant 1/2 never equals zero, so there's no restriction. Always simplify first, then determine domain from the simplified form.

Practice Quiz

Test your knowledge with interactive questions

$2x+\frac{6}{x}=18$

What is the domain of the above equation?

FAQ

Everything you need to know about this question

Why isn't x ≠ 1/2 part of the domain restriction?

Great question! The denominator is the constant 1/2, not the variable x. Since 1/2 ≠ 0, there's no division by zero issue. Domain restrictions only occur when the variable could make a denominator zero.

How do I know when to simplify before finding the domain?

Always simplify first! The original form might look complicated, but once you simplify $\frac{10x}{\frac{1}{2}}$ to 20x, you can clearly see it's just a linear function with no restrictions.

What if the denominator had x in it instead of 1/2?

Then you'd need to find where the denominator equals zero! For example, if you had $\frac{10x}{x-2}$ , then x ≠ 2 because that would make the denominator zero.

How do I remember the rule for dividing by fractions?

Think "flip and multiply"! Dividing by $\frac{1}{2}$ is the same as multiplying by $\frac{2}{1} = 2$ . So $\frac{10x}{\frac{1}{2}} = 10x \times 2$ .

Can a linear function ever have domain restrictions?

In standard form like f(x) = mx + b, linear functions have all real numbers as their domain. Domain restrictions only appear in rational, radical, or logarithmic functions where certain x-values cause undefined operations.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Functions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime