Determine the Domain: Analyzing 24/(21x-7)

Q: Why can't the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the function has no meaningful value, so we must exclude those x-values from the domain.

Q: What does 'x ≠ 1/3' mean exactly?

This means x can be any real number except 1/3. So x could be 0, 2, -5, 1/2, etc., but never exactly 1/3 because that makes the denominator zero.

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

The domain is $ (-\infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty) $. The union symbol ∪ connects the two intervals, excluding the point x = 1/3.

Q: What if I got a different answer when solving 21x - 7 = 0?

Double-check your algebra! Add 7 to both sides: $ 21x = 7 $, then divide by 21: $ x = \frac{7}{21} = \frac{1}{3} $. Always simplify fractions!

Q: Do I need to consider the numerator for the domain?

No! The numerator (24) is just a constant and doesn't affect the domain. Only worry about when the denominator equals zero for rational functions.

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

The domain is all possible x-values (inputs), while the range is all possible y-values (outputs). For domain, we only care about what x-values make the function undefined.

Rational Function Domains with Linear Denominators

Given the following function:

$\frac{24}{21x-7}$

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 yes, what is it?

00:03 Does the function have a domain? If yes, what is it?

00:07 So let's see what solution zeros the denominator

00:11 Let's isolate X

00:25 Let's factorize 21 into factors 7 and 3

00:29 Let's simplify what we can

00:32 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:

$\frac{24}{21x-7}$

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{24}{21x-7}$ , we need to ensure that the denominator is not equal to zero.

Step 1: Set the denominator equal to zero and solve for $x$ :

$21x - 7 = 0$
$21x = 7$
$x = \frac{7}{21}$
$x = \frac{1}{3}$

The function is undefined when $x = \frac{1}{3}$ because it would cause division by zero.

Step 2: The domain of the function is all real numbers except $x = \frac{1}{3}$ .

Therefore, the domain of the function is all $x$ such that $x \neq \frac{1}{3}$ .

Thus, the correct answer is $\boxed{ x\ne\frac{1}{3}}$ .

Final Answer

$x\ne\frac{1}{3}$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Exclude all values where denominator equals zero
Technique: Set 21x - 7 = 0, solve to get x = 1/3
Check: Substitute x = 1/3: 21(1/3) - 7 = 0, undefined ✓

Common Mistakes

Avoid these frequent errors

Setting the entire function equal to zero instead of just the denominator
Don't solve 24/(21x-7) = 0 to find domain restrictions! This finds where the function equals zero, not where it's undefined. Always set only the denominator equal to zero: 21x - 7 = 0.

Practice Quiz

Test your knowledge with interactive questions

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

$\frac{9x}{4}$

FAQ

Everything you need to know about this question

Why can't the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the function has no meaningful value, so we must exclude those x-values from the domain.

What does 'x ≠ 1/3' mean exactly?

This means x can be any real number except 1/3. So x could be 0, 2, -5, 1/2, etc., but never exactly 1/3 because that makes the denominator zero.

How do I write the domain in interval notation?

The domain is $(-\infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$ . The union symbol ∪ connects the two intervals, excluding the point x = 1/3.

What if I got a different answer when solving 21x - 7 = 0?

Double-check your algebra! Add 7 to both sides: $21x = 7$ , then divide by 21: $x = \frac{7}{21} = \frac{1}{3}$ . Always simplify fractions!

Do I need to consider the numerator for the domain?

No! The numerator (24) is just a constant and doesn't affect the domain. Only worry about when the denominator equals zero for rational functions.

What's the difference between domain and range?

The domain is all possible x-values (inputs), while the range is all possible y-values (outputs). For domain, we only care about what x-values make the function undefined.

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