Determine the Domain of the Rational Function 20/(10x-5)

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, creating a vertical asymptote on the graph.

Q: What does the domain notation x ≠ 1/2 mean?

This means x can be any real number except 1/2. In interval notation, this would be $ (-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty) $.

Q: How do I solve 10x - 5 = 0 step by step?

Step 1: Add 5 to both sides: 10x = 5Step 2: Divide by 10: $ x = \frac{5}{10} = \frac{1}{2} $

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

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

Q: Can the domain ever be all real numbers for a rational function?

Only if the denominator never equals zero. For example, $ \frac{1}{x^2+1} $ has domain all reals because $ x^2+1 $ is always positive.

Domain Finding with Rational Denominators

Look at the following function:

$\frac{20}{10x-5}$

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

00:07 Does the function have a domain? Let's find out together.

00:11 To find the domain, remember, we can't divide by zero.

00:16 So, let's figure out what value makes the denominator zero.

00:20 Next, we'll isolate X to find the solution.

00:32 Let's break down 10 into factors, 5 and 2.

00:36 Now, we'll simplify as much as we can.

00:40 And there you have it! That's our solution.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Look at the following function:

$\frac{20}{10x-5}$

What is the domain of the function?

Step-by-step solution

To determine the domain of the function $\frac{20}{10x-5}$ , we need to ensure that the denominator is not zero.

Step 1: Identify the denominator, which is $10x - 5$ .
Step 2: Set the denominator equal to zero and solve for $x$ . This gives us the equation:

$10x - 5 = 0$

Step 3: Add 5 to both sides of the equation:

$10x = 5$

Step 4: Divide both sides by 10 to isolate $x$ :

$x = \frac{5}{10}$

Step 5: Simplify the fraction:

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

This means that the function is undefined at $x = \frac{1}{2}$ . Therefore, the domain of the function is all real numbers except $x = \frac{1}{2}$ .

Therefore, the domain of the function 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: Rational functions are undefined where denominators equal zero
Technique: Set 10x - 5 = 0, then solve to get x = 1/2
Check: Substitute x = 1/2: 10(1/2) - 5 = 0, confirming exclusion ✓

Common Mistakes

Avoid these frequent errors

Setting the numerator equal to zero instead
Don't set 20 = 0 to find restrictions = no solution found! The numerator being zero doesn't create undefined values, only makes the function equal zero. Always set the denominator equal to zero to find domain restrictions.

Practice Quiz

Test your knowledge with interactive questions

$2x+\frac{6}{x}=18$

What is the domain of the above 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 function has no value at that point, creating a vertical asymptote on the graph.

What does the domain notation x ≠ 1/2 mean?

This means x can be any real number except 1/2. In interval notation, this would be $(-\infty, \frac{1}{2}) \cup (\frac{1}{2}, \infty)$ .

How do I solve 10x - 5 = 0 step by step?

Step 1: Add 5 to both sides: 10x = 5
Step 2: Divide by 10: $x = \frac{5}{10} = \frac{1}{2}$

What if there are multiple terms in the denominator?

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

Can the domain ever be all real numbers for a rational function?

Only if the denominator never equals zero. For example, $\frac{1}{x^2+1}$ has domain all reals because $x^2+1$ is always positive.

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