Uncover the Domain: Analyze the Rational Expression (3x+4)/(1/2x)

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

When the numerator equals zero, the function value is zero (which is perfectly fine). Only when the denominator equals zero do we get undefined values that must be excluded from the domain.

Q: How do I handle fractions in the denominator like 1/2x?

Treat $ \frac{1}{2}x $ as a single expression. Set it equal to zero: $ \frac{1}{2}x = 0 $. Since any number times zero equals zero, we get x = 0.

Q: What's the difference between x ≠ 0 and x ≠ 1/2?

$ x \neq 0 $ means x cannot be zero. $ x \neq \frac{1}{2} $ means x cannot be one-half. In our problem, only zero makes the denominator undefined.

Q: Can I simplify the fraction first before finding the domain?

Be careful! You can rewrite $ \frac{3x+4}{\frac{1}{2}x} $ as $ \frac{2(3x+4)}{x} $, but the domain restriction x ≠ 0 remains the same.

Q: How do I write the final domain answer?

You can write it as: $ x \neq 0 $, or "all real numbers except x = 0", or in interval notation: $ (-\infty, 0) \cup (0, \infty) $.

Rational Function Domains with Fractional Denominators

Given the following function:

$\frac{3x+4}{\frac{1}{2}x}$

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

00:06 Therefore let's see what solution makes the denominator zero

00:11 Let's isolate X

00:14 0 divided by any number is always equal to 0

00:18 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:

$\frac{3x+4}{\frac{1}{2}x}$

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{3x+4}{\frac{1}{2}x}$ , we need to ensure that the denominator does not equal zero, as division by zero is undefined.

First, identify the denominator in the function, which is $\frac{1}{2}x$ .

Set the denominator equal to zero to find the values that make it undefined:
$\frac{1}{2}x = 0$

To solve for $x$ , multiply both sides by 2 to clear the fraction:
$x = 0$

The domain excludes the value $x = 0$ because it would make the denominator zero, rendering the function undefined.

Thus, the domain of the function $\frac{3x+4}{\frac{1}{2}x}$ is all real numbers except $x = 0$ .

Therefore, the correct answer is $x \neq 0$ .

Final Answer

$x\ne0$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Exclude values that make any denominator equal zero
Technique: Set $\frac{1}{2}x = 0$ and solve: multiply by 2 gives x = 0
Check: Substitute x = 0: denominator becomes 0, confirming undefined ✓

Common Mistakes

Avoid these frequent errors

Confusing numerator and denominator restrictions
Don't set the numerator 3x+4 = 0 to find domain restrictions = wrong exclusions! The numerator can equal zero (creating zeros of the function), but only denominators create domain restrictions. Always focus on when denominators equal zero.

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

When the numerator equals zero, the function value is zero (which is perfectly fine). Only when the denominator equals zero do we get undefined values that must be excluded from the domain.

How do I handle fractions in the denominator like 1/2x?

Treat $\frac{1}{2}x$ as a single expression. Set it equal to zero: $\frac{1}{2}x = 0$ . Since any number times zero equals zero, we get x = 0.

What's the difference between x ≠ 0 and x ≠ 1/2?

$x \neq 0$ means x cannot be zero. $x \neq \frac{1}{2}$ means x cannot be one-half. In our problem, only zero makes the denominator undefined.

Can I simplify the fraction first before finding the domain?

Be careful! You can rewrite $\frac{3x+4}{\frac{1}{2}x}$ as $\frac{2(3x+4)}{x}$ , but the domain restriction x ≠ 0 remains the same.

How do I write the final domain answer?

You can write it as: $x \neq 0$ , or "all real numbers except x = 0", or in interval notation: $(-\infty, 0) \cup (0, \infty)$ .

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