Identify the Domain: Uncovering Restrictions in 3x+4 over 2x-1

Q: Why can't the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the fraction has no meaningful value, so these x-values must be excluded from the domain.

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

Domain is all possible x-values (input), while range is all possible y-values (output). For rational functions, focus on denominator restrictions for domain.

Q: Do I need to worry about the numerator being zero?

No! When the numerator is zero, the function equals zero, which is perfectly fine. Only worry about the denominator being zero.

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

For this function, the domain is $ (-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty) $. The union symbol ∪ connects the two intervals that exclude x = 1/2.

Rational Functions with Denominator Restrictions

Consider the following function:

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

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? And if so, what is it?

00:03 To find the domain, remember that division by 0 is not allowed

00:06 So let's see what solution zeros the denominator

00:10 Let's isolate X

00:23 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

Consider the following function:

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

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{3x+4}{2x-1}$ , follow these steps:

Step 1: Identify the denominator of the rational function, which is $2x-1$ .
Step 2: Set the denominator equal to zero to find the values of $x$ that make the function undefined:
$2x - 1 = 0$ .
Step 3: Solve for $x$ :
Add 1 to both sides: $2x = 1$ .
Divide both sides by 2: $x = \frac{1}{2}$ .

The value $x = \frac{1}{2}$ makes the denominator zero, which means the function $\frac{3x+4}{2x-1}$ is undefined at $x = \frac{1}{2}$ . Therefore, this value must be excluded from the domain.

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

Therefore, the solution to the problem is $x \ne \frac{1}{2}$ .

Final Answer

$x\ne\frac{1}{2}$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Exclude values that make the denominator equal zero
Method: Set denominator 2x - 1 = 0, solve to get x = 1/2
Verification: Check that 2(1/2) - 1 = 0 confirms restriction at x = 1/2 ✓

Common Mistakes

Avoid these frequent errors

Setting the entire function equal to zero instead of just the denominator
Don't solve 3x+4/2x-1 = 0 to find domain restrictions! This finds zeros of the function, not domain restrictions. Always set only the denominator equal to zero: 2x - 1 = 0.

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 the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the fraction has no meaningful value, so these x-values must be excluded from the domain.

What's the difference between domain and range?

Domain is all possible x-values (input), while range is all possible y-values (output). For rational functions, focus on denominator restrictions for domain.

Do I need to worry about the numerator being zero?

No! When the numerator is zero, the function equals zero, which is perfectly fine. Only worry about the denominator being zero.

How do I write the domain in interval notation?

For this function, the domain is $(-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$ . The union symbol ∪ connects the two intervals that exclude x = 1/2.

What if the denominator is a more complex expression?

The same rule applies! Set the denominator equal to zero and solve completely. You might get multiple restrictions if the denominator factors into several terms.

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