Find the Domain of (5+4x)/(2+x²): Rational Function Analysis

Q: Why doesn't this rational function have any restrictions?

The denominator is $ 2+x^2 $. Since squares are always non-negative ($ x^2 \geq 0 $), we have $ 2+x^2 \geq 2 $, which is always positive!

Q: How do I know when a denominator can never be zero?

Look for patterns like positive constant + square term. Since $ x^2 \geq 0 $ for all real numbers, adding a positive number guarantees the result is always positive.

Q: What if the denominator was just x²?

Then $ x^2 = 0 $ when $ x = 0 $, so the domain would be all real numbers except x = 0. The "+2" in our problem prevents this restriction.

Q: Do I still need to check even if it looks obvious?

Yes! Always solve the equation denominator = 0 systematically. This builds good habits and prevents mistakes on more complex problems.

Q: What does 'entire domain' or 'all real numbers' mean?

It means you can substitute any real number for x and the function will have a valid output. There are no values that make the function undefined.

Rational Function Domains with Always-Positive Denominators

Look at the following function:

$\frac{5+4x}{2+x^2}$

Does the function have a domain? If so, what is it?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:11 First, we need to know if the function has a domain. And if it does, what is it?

00:16 To find the domain, keep in mind that dividing by zero is not allowed.

00:21 So, let's figure out what value makes the denominator equal to zero.

00:26 Alright, let's try to isolate X to find that value.

00:31 Remember, any number squared is always greater than zero. That's positive.

00:36 And there you have it! That's the solution to our problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$\frac{5+4x}{2+x^2}$

Does the function have a domain? If so, what is it?

Step-by-step solution

Since the denominator is positive for all $x$ , the domain of the function is the entire domain.

That is, all values of $x$ . Therefore, there is no domain limits.

Final Answer

No, the entire domain

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Find where denominator equals zero to identify restrictions
Analysis: Set $2+x^2 = 0$ gives $x^2 = -2$
Verification: Since $x^2 \geq 0$ always, $2+x^2 \geq 2 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Assuming all rational functions have domain restrictions
Don't automatically exclude values without checking if the denominator can actually equal zero! This leads to unnecessarily restricted domains. Always solve the equation denominator = 0 first to see if real solutions exist.

Practice Quiz

Test your knowledge with interactive questions

Given the following function:

$\frac{5-x}{2-x}$

Does the function have a domain? If so, what is it?

FAQ

Everything you need to know about this question

Why doesn't this rational function have any restrictions?

The denominator is $2+x^2$ . Since squares are always non-negative ( $x^2 \geq 0$ ), we have $2+x^2 \geq 2$ , which is always positive!

How do I know when a denominator can never be zero?

Look for patterns like positive constant + square term. Since $x^2 \geq 0$ for all real numbers, adding a positive number guarantees the result is always positive.

What if the denominator was just x²?

Then $x^2 = 0$ when $x = 0$ , so the domain would be all real numbers except x = 0. The "+2" in our problem prevents this restriction.

Do I still need to check even if it looks obvious?

Yes! Always solve the equation denominator = 0 systematically. This builds good habits and prevents mistakes on more complex problems.

What does 'entire domain' or 'all real numbers' mean?

It means you can substitute any real number for x and the function will have a valid output. There are no values that make the function undefined.

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Find the Domain of (5+4x)/(2+x²): Rational Function Analysis

Rational Function Domains with Always-Positive Denominators

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why doesn't this rational function have any restrictions?

How do I know when a denominator can never be zero?

What if the denominator was just x²?

Do I still need to check even if it looks obvious?

What does 'entire domain' or 'all real numbers' mean?

🌟 Unlock Your Math Potential

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