Analyzing the Rational Function: Determine the Domain of (5-x)/(2-x)

Q: Why don't we worry about when the numerator equals zero?

When the numerator equals zero, the function just equals zero, which is perfectly valid! Only when the denominator equals zero do we get undefined values that must be excluded from the domain.

Q: What does x ≠ 2 actually mean?

It means x can be any real number except 2. So x could be 1.9, 2.1, -5, 100, or any other number, just not exactly 2.

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

The domain $ x \ne 2 $ in interval notation is (-∞, 2) ∪ (2, ∞). This shows all real numbers except the gap at x = 2.

Q: What happens if I try to substitute x = 2?

You get $ \frac{5-2}{2-2} = \frac{3}{0} $, which is undefined. Division by zero is impossible in mathematics!

Q: Could there be more than one restriction?

Yes! If the denominator had multiple factors like (x-2)(x+1), then both x = 2 and x = -1 would be excluded from the domain.

Rational Function Domain with Denominator Restrictions

Given the following function:

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

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

❤️ 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?

00:04 To find the domain, remember that division by 0 is not allowed

00:08 Therefore, let's see what solution zeroes the denominator

00:10 Let's isolate X

00:17 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

Given the following function:

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

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

Step-by-step solution

To determine the domain of the function $\frac{5-x}{2-x}$ , we need to identify and exclude any values of $x$ that make the function undefined. This occurs when the denominator equals zero.

Step 1: Set the denominator equal to zero:
$2-x = 0$
Step 2: Solve for $x$ :
Adding $x$ to both sides gives $2 = x$ . Hence, $x = 2$ .

This means that the function is undefined when $x = 2$ . Therefore, the domain of the function consists of all real numbers except $x = 2$ .

Thus, the domain is: $x \ne 2$ .

The correct answer choice is:

Yes, $x\ne2$

Final Answer

Yes, $x\ne2$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Rational functions are undefined when denominators equal zero
Technique: Set denominator 2-x = 0, solve to get x = 2
Check: Substitute x = 2: denominator becomes 2-2 = 0, confirming undefined ✓

Common Mistakes

Avoid these frequent errors

Checking the numerator instead of denominator
Don't set the numerator 5-x = 0 to find restrictions = wrong domain! The numerator being zero just makes the function equal zero, not undefined. Always check only the denominator for domain restrictions.

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 don't we worry about when the numerator equals zero?

When the numerator equals zero, the function just equals zero, which is perfectly valid! Only when the denominator equals zero do we get undefined values that must be excluded from the domain.

What does x ≠ 2 actually mean?

It means x can be any real number except 2. So x could be 1.9, 2.1, -5, 100, or any other number, just not exactly 2.

How do I write the domain in interval notation?

The domain $x \ne 2$ in interval notation is (-∞, 2) ∪ (2, ∞). This shows all real numbers except the gap at x = 2.

What happens if I try to substitute x = 2?

You get $\frac{5-2}{2-2} = \frac{3}{0}$ , which is undefined. Division by zero is impossible in mathematics!

Could there be more than one restriction?

Yes! If the denominator had multiple factors like (x-2)(x+1), then both x = 2 and x = -1 would be excluded from the domain.

🌟 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