Determine the Domain of the Rational Function: (5x+2)/(4x-3)

Q: Why can't the denominator be zero?

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

Q: What if the numerator is also zero when x = 3/4?

Even if both numerator and denominator are zero, we still exclude that x-value from the domain. The function remains undefined at that point regardless of the numerator.

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

The domain is $ (-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty) $. The union symbol ∪ connects the two intervals that exclude $ x = \frac{3}{4} $.

Q: What if there are multiple terms in the denominator?

Set the entire denominator equal to zero and solve. Whether it's 4x - 3 or a more complex expression, the process is the same: find all values that make the bottom zero.

Q: Is there a difference between 'domain' and 'range' here?

Domain is the set of allowed x-values (input). Range is the set of possible y-values (output). For rational functions, focus on domain first by finding denominator restrictions.

Rational Function Domains with Zero Denominators

Look at the following function:

$\frac{5x+2}{4x-3}$

What is the domain of the function?

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

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

00:07 Therefore, let's find the solution that makes the denominator zero

00:10 Let's isolate X

00:21 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:

$\frac{5x+2}{4x-3}$

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{5x+2}{4x-3}$ , we must identify the values of $x$ that make the denominator zero, as these values are not allowed in the domain of a rational function.

Step 1: Set the denominator equal to zero:

$4x - 3 = 0$

Step 2: Solve for $x$ :

$4x = 3$

$x = \frac{3}{4}$

The function is undefined at $x = \frac{3}{4}$ because division by zero is not permissible.

Therefore, the domain of the function is all real numbers except $x = \frac{3}{4}$ . This can be expressed as:

$x \ne \frac{3}{4}$

The correct answer, based on the choices given, is:

$x \ne \frac{3}{4}$

Final Answer

$x\ne\frac{3}{4}$

Key Points to Remember

Essential concepts to master this topic

Rule: Set denominator equal to zero and solve for restricted values
Technique: From 4x - 3 = 0, add 3 then divide by 4
Check: Substitute $x = \frac{3}{4}$ into denominator: 4(3/4) - 3 = 0 ✓

Common Mistakes

Avoid these frequent errors

Finding where the numerator equals zero instead of denominator
Don't set the numerator 5x + 2 = 0 to find domain restrictions = wrong excluded value! 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

$\frac{6}{x+5}=1$

What is the field of application of the equation?

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 meaning, so we must exclude those x-values from the domain.

What if the numerator is also zero when x = 3/4?

Even if both numerator and denominator are zero, we still exclude that x-value from the domain. The function remains undefined at that point regardless of the numerator.

How do I write the domain in interval notation?

The domain is $(-\infty, \frac{3}{4}) \cup (\frac{3}{4}, \infty)$ . The union symbol ∪ connects the two intervals that exclude $x = \frac{3}{4}$ .

What if there are multiple terms in the denominator?

Set the entire denominator equal to zero and solve. Whether it's 4x - 3 or a more complex expression, the process is the same: find all values that make the bottom zero.

Is there a difference between 'domain' and 'range' here?

Domain is the set of allowed x-values (input). Range is the set of possible y-values (output). For rational functions, focus on domain first by finding denominator restrictions.

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Functions questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime