Determining the Domain of the Rational Function: 5+4x/x²

Q: Why doesn't the numerator 5+4x affect the domain?

The numerator can equal zero without problems - it just makes the whole fraction equal zero. Only when the denominator equals zero do we get undefined values that must be excluded from the domain.

Q: What does x ≠ 0 mean exactly?

It means x can be any real number except 0. You can use positive numbers, negative numbers, fractions, decimals - just not zero itself because that makes the denominator zero.

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

The domain $ x \ne 0 $ in interval notation is (-∞, 0) ∪ (0, ∞). This shows all real numbers except zero using two separate intervals.

Q: What happens if I accidentally substitute x = 0?

You'll get $ \frac{5+4(0)}{0^2} = \frac{5}{0} $, which is undefined. Division by zero is impossible in mathematics, which is exactly why we exclude x = 0 from the domain!

Q: Are there other values I should check besides x = 0?

No! Since the denominator is $ x^2 $, the only solution to $ x^2 = 0 $ is x = 0. Always solve the denominator equation completely to find all restrictions.

Rational Function Domains with Zero Denominators

Given the following function:

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

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

00:09 Does the function have a domain? If it does, let's find out what it is.

00:14 To find the domain, remember, we cannot divide by zero.

00:19 So, let's find what makes the denominator zero.

00:22 First, we need to isolate X.

00:25 And that's 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:

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

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

Step-by-step solution

To determine the domain of the function $\frac{5+4x}{x^2}$ , we need to identify values of $x$ that cause the denominator to be zero, as the function is undefined for these values.

Step 1: Set the denominator equal to zero:

$x^2 = 0$

Step 2: Solve for $x$ :

Taking the square root of both sides gives $x = 0$ .

The function is undefined at $x = 0$ , so we must exclude this value from the domain.

Thus, the domain of the function is all real numbers except $x = 0$ .

The domain can be expressed as: $x \ne 0$ .

Therefore, the correct answer is option 3: Yes, $x \ne 0$ .

Final Answer

Yes, $x\ne0$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Function undefined when denominator equals zero
Technique: Set $x^2 = 0$ and solve to get $x = 0$
Check: Verify $\frac{5+4(0)}{0^2}$ is undefined (division by zero) ✓

Common Mistakes

Avoid these frequent errors

Focusing on the numerator instead of denominator
Don't set the numerator 5+4x equal to zero to find domain restrictions = wrong exclusions! The numerator being zero just makes the function equal zero, not undefined. Always set only the denominator equal to zero to find domain restrictions.

Practice Quiz

Test your knowledge with interactive questions

$22(\frac{2}{x}-1)=30$

What is the domain of the equation above?

FAQ

Everything you need to know about this question

Why doesn't the numerator 5+4x affect the domain?

The numerator can equal zero without problems - it just makes the whole fraction equal zero. Only when the denominator equals zero do we get undefined values that must be excluded from the domain.

What does x ≠ 0 mean exactly?

It means x can be any real number except 0. You can use positive numbers, negative numbers, fractions, decimals - just not zero itself because that makes the denominator zero.

How do I write the domain in interval notation?

The domain $x \ne 0$ in interval notation is (-∞, 0) ∪ (0, ∞). This shows all real numbers except zero using two separate intervals.

What happens if I accidentally substitute x = 0?

You'll get $\frac{5+4(0)}{0^2} = \frac{5}{0}$ , which is undefined. Division by zero is impossible in mathematics, which is exactly why we exclude x = 0 from the domain!

Are there other values I should check besides x = 0?

No! Since the denominator is $x^2$ , the only solution to $x^2 = 0$ is x = 0. Always solve the denominator equation completely to find all restrictions.

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