Identify the Domain of the Function: Issues with 1/(5x - 4)

Q: Why can't the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the fraction $ \frac{1}{5x-4} $ has no meaning, so these x-values must be excluded from the domain.

Q: How do I write the domain properly?

You can write it as $ x \ne \frac{4}{5} $ or as interval notation: $ (-\infty, \frac{4}{5}) \cup (\frac{4}{5}, \infty) $. Both show that x cannot equal 4/5.

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

Set the entire denominator equal to zero and solve. For example, with $ \frac{1}{x^2-4} $, solve $ x^2-4=0 $ to get x = ±2.

Q: Can the domain ever include all real numbers?

Yes! If the denominator is never zero (like $ \frac{1}{x^2+1} $), then the domain includes all real numbers since $ x^2+1 $ is always positive.

Q: Do I need to simplify fractions in my final answer?

Always simplify! If you get $ x = \frac{8}{10} $, reduce it to $ x = \frac{4}{5} $. This matches the standard form and prevents confusion.

Function Domains with Rational Expressions

Look the following function:

$\frac{1}{5x-4}$

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?

00:04 To find the domain, remember that division by 0 is not allowed

00:07 So let's see what solution zeros the denominator

00:13 Let's isolate X

00:31 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 the following function:

$\frac{1}{5x-4}$

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{1}{5x-4}$ , we need to find the values of $x$ for which the function is undefined. This occurs when the denominator equals zero:

First, set the denominator equal to zero:
$5x - 4 = 0$

Next, solve for $x$ :
$5x = 4$
$x = \frac{4}{5}$

The function is undefined at $x = \frac{4}{5}$ . Therefore, the domain of the function includes all real numbers except $x = \frac{4}{5}$ .

In mathematical notation, the domain is:
$x \ne \frac{4}{5}$ .

This matches choice 3 among the given options.

Final Answer

$x\ne\frac{4}{5}$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Function undefined when denominator equals zero
Technique: Set 5x - 4 = 0, solve: x = 4/5
Check: Substitute back: 5(4/5) - 4 = 4 - 4 = 0 ✓

Common Mistakes

Avoid these frequent errors

Setting the numerator equal to zero instead of denominator
Don't set the numerator (1) equal to zero = wrong restriction! The numerator being zero makes the function equal zero, not undefined. Always set the denominator equal to zero to find where the function is undefined.

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 $\frac{1}{5x-4}$ has no meaning, so these x-values must be excluded from the domain.

How do I write the domain properly?

You can write it as $x \ne \frac{4}{5}$ or as interval notation: $(-\infty, \frac{4}{5}) \cup (\frac{4}{5}, \infty)$ . Both show that x cannot equal 4/5.

What if there are multiple terms in the denominator?

Set the entire denominator equal to zero and solve. For example, with $\frac{1}{x^2-4}$ , solve $x^2-4=0$ to get x = ±2.

Can the domain ever include all real numbers?

Yes! If the denominator is never zero (like $\frac{1}{x^2+1}$ ), then the domain includes all real numbers since $x^2+1$ is always positive.

Do I need to simplify fractions in my final answer?

Always simplify! If you get $x = \frac{8}{10}$ , reduce it to $x = \frac{4}{5}$ . This matches the standard form and prevents confusion.

🌟 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