Finding the Domain of the Function: 23/(5x-2)

Q: Why can't the denominator equal zero?

Division by zero is undefined in mathematics! When $ 5x - 2 = 0 $, we're trying to divide 23 by 0, which has no mathematical meaning.

Q: What does the domain notation mean?

The notation $ x \ne \frac{2}{5} $ means 'x is not equal to 2/5'. This tells us that all real numbers except 2/5 are allowed as inputs.

Q: How do I solve 5x - 2 = 0?

Add 2 to both sides: $ 5x = 2 $Then divide by 5: $ x = \frac{2}{5} $This is the forbidden value for our domain!

Q: Can the domain ever include the excluded value?

Never! Once a value makes the denominator zero, it's permanently excluded from the domain. There's no way to 'fix' division by zero.

Rational Function Domains with Linear Denominators

Given the following function:

$\frac{23}{5x-2}$

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

00:00 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

00:08 Therefore, let's find the solution that makes the denominator zero

00:15 Let's isolate X

00:27 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:

$\frac{23}{5x-2}$

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

Step-by-step solution

To determine the domain of the function $\frac{23}{5x-2}$ , follow these steps:

Step 1: Identify where the function is undefined by setting the denominator equal to zero.
Equation: $5x - 2 = 0$
Step 2: Solve the equation for $x$ .

Let's perform the calculation:
Step 1: Set $5x - 2 = 0$ .

Step 2: Solve for $x$ by adding 2 to both sides:
$5x = 2$

Next, divide both sides by 5:
$x = \frac{2}{5}$

This shows that the function is undefined at $x = \frac{2}{5}$ , thus excluding this point from the domain of the function.

The domain of $\frac{23}{5x-2}$ consists of all real numbers except $x = \frac{2}{5}$ .

Therefore, the domain is expressed as $x \ne \frac{2}{5}$ .

Considering the multiple-choice options, the correct choice is:

Yes, $x\ne\frac{2}{5}$

Final Answer

Yes, $x\ne\frac{2}{5}$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Exclude values that make denominators equal zero
Technique: Set denominator $5x - 2 = 0$ and solve for $x = \frac{2}{5}$
Check: Substitute excluded value: $5(\frac{2}{5}) - 2 = 0$ confirms undefined point ✓

Common Mistakes

Avoid these frequent errors

Setting the numerator equal to zero instead of denominator
Don't set 23 = 0 to find domain restrictions = no solution found! The numerator being zero creates zeros of the function, not domain restrictions. Always set only 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 equal zero?

Division by zero is undefined in mathematics! When $5x - 2 = 0$ , we're trying to divide 23 by 0, which has no mathematical meaning.

What does the domain notation mean?

The notation $x \ne \frac{2}{5}$ means 'x is not equal to 2/5'. This tells us that all real numbers except 2/5 are allowed as inputs.

How do I solve 5x - 2 = 0?

Add 2 to both sides: $5x = 2$
Then divide by 5: $x = \frac{2}{5}$
This is the forbidden value for our domain!

What if I had multiple terms in the denominator?

The same process applies! Set the entire denominator equal to zero and solve. For example, with $\frac{1}{(x-3)(x+1)}$ , you'd exclude both x = 3 and x = -1.

Can the domain ever include the excluded value?

Never! Once a value makes the denominator zero, it's permanently excluded from the domain. There's no way to 'fix' division by zero.

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