Determine the Domain: Tackle (10x-3)/(5x-3) in Rational Function Analysis

Q: Why can't the denominator be zero?

Division by zero is undefined in mathematics! When the denominator equals zero, the function has no value at that point, so we must exclude it from the domain.

Q: What happens if the numerator is zero?

If only the numerator is zero, the function simply equals zero at that point. This is perfectly fine and doesn't affect the domain - only denominators being zero matter!

Q: How do I write domain restrictions properly?

Use the notation $ x \neq \frac{3}{5} $ to show that x cannot equal 3/5. You can also write it as "all real numbers except x = 3/5".

Q: Can a rational function have multiple domain restrictions?

Yes! If the denominator has multiple factors, each factor that equals zero creates a separate restriction. For example, $ \frac{1}{(x-2)(x+1)} $ excludes both x = 2 and x = -1.

Domain Restrictions with Rational Functions

Look at the following function:

$\frac{10x-3}{5x-3}$

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

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

00:11 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

Look at the following function:

$\frac{10x-3}{5x-3}$

What is the domain of the function?

Step-by-step solution

To find the domain of the function $\frac{10x-3}{5x-3}$ , we'll follow these steps:

Identify the denominator: $B(x) = 5x - 3$ .
Set the denominator equal to zero: $5x - 3 = 0$ .
Solve for $x$ : Add 3 to both sides, getting $5x = 3$ . Then, divide by 5: $x = \frac{3}{5}$ .
Conclude that the domain is all real numbers except $x = \frac{3}{5}$ , since this makes the denominator zero.

Therefore, the domain of the function is all real numbers except $x\ne\frac{3}{5}$ .

Final Answer

$x\ne\frac{3}{5}$

Key Points to Remember

Essential concepts to master this topic

Rule: Domain excludes all values that make denominator zero
Technique: Set denominator 5x - 3 = 0, solve x = 3/5
Check: Substitute x = 3/5: denominator becomes 5(3/5) - 3 = 0 ✓

Common Mistakes

Avoid these frequent errors

Setting the numerator equal to zero instead of denominator
Don't solve 10x - 3 = 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

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 function has no value at that point, so we must exclude it from the domain.

What happens if the numerator is zero?

If only the numerator is zero, the function simply equals zero at that point. This is perfectly fine and doesn't affect the domain - only denominators being zero matter!

How do I write domain restrictions properly?

Use the notation $x \neq \frac{3}{5}$ to show that x cannot equal 3/5. You can also write it as "all real numbers except x = 3/5".

Can a rational function have multiple domain restrictions?

Yes! If the denominator has multiple factors, each factor that equals zero creates a separate restriction. For example, $\frac{1}{(x-2)(x+1)}$ excludes both x = 2 and x = -1.

Do I need to simplify the function first?

Be careful! If you simplify by canceling factors, you might hide domain restrictions. Always find the domain using the original form before any simplification.

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