Finding the Domain of 5/x: Analyzing Function Restrictions

Q: Why can't I divide by zero?

Division by zero is undefined in mathematics! Think of it this way: $ \frac{5}{0} $ would mean "how many groups of 0 make 5?" - which has no meaningful answer.

Q: What does the domain notation x ≠ 0 mean?

The notation $ x \ne 0 $ means "x cannot equal zero." The domain includes all real numbers except 0 - so x can be any positive or negative number, just not zero.

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

For $ \frac{5}{x} $, the domain is (-∞, 0) ∪ (0, ∞). This shows all real numbers from negative infinity to positive infinity, with a gap at zero.

Q: What if the denominator was x - 5 instead?

Then you'd set $ x - 5 = 0 $, so $ x = 5 $. The domain would be all real numbers except 5, written as $ x \ne 5 $.

Q: Can the numerator affect the domain?

Usually not! The numerator can be any value, including zero. Only the denominator creates domain restrictions because division by zero is undefined.

Function Domains with Division Restrictions

Look at the following function:

$\frac{5}{x}$

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

00:00 Does the function have a domain? And if so, what is it?

00:03 To find the domain, remember that we cannot divide by 0

00:07 Therefore, let's see what solution makes the denominator zero

00:10 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:

$\frac{5}{x}$

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

Step-by-step solution

Since the unknown variable is in the denominator, we should remember that the denominator cannot be equal to 0.

In other words, $x\ne0$

The domain of the function is all those values that, when substituted into the function, will make the function legal and defined.

The domain in this case will be all real numbers that are not equal to 0.

Final Answer

Yes, $x\ne0$

Key Points to Remember

Essential concepts to master this topic

Division Rule: The denominator in any fraction cannot equal zero
Technique: Set denominator equal to zero: x = 0
Check: Domain excludes x = 0, so $\frac{5}{x}$ undefined when x = 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing numerator with denominator restrictions
Don't set the numerator 5 equal to zero = wrong restriction! The numerator can be any value. Always focus on the denominator - only when the denominator equals zero does the function become undefined.

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 can't I divide by zero?

Division by zero is undefined in mathematics! Think of it this way: $\frac{5}{0}$ would mean "how many groups of 0 make 5?" - which has no meaningful answer.

What does the domain notation x ≠ 0 mean?

The notation $x \ne 0$ means "x cannot equal zero." The domain includes all real numbers except 0 - so x can be any positive or negative number, just not zero.

How do I write the domain in interval notation?

For $\frac{5}{x}$ , the domain is (-∞, 0) ∪ (0, ∞). This shows all real numbers from negative infinity to positive infinity, with a gap at zero.

What if the denominator was x - 5 instead?

Then you'd set $x - 5 = 0$ , so $x = 5$ . The domain would be all real numbers except 5, written as $x \ne 5$ .

Can the numerator affect the domain?

Usually not! The numerator can be any value, including zero. Only the denominator creates domain restrictions because division by zero is undefined.

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