Analyzing the Domain of 2x+2/9x+6: Understanding the Function's Limits

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 or hole in the graph.

Q: What does x ≠ -2/3 actually mean?

This notation means x can be any real number except -2/3. So x could be -1, 0, 5, or even -0.66, but never exactly -2/3.

Q: Do I need to simplify the function first?

Not for finding the domain! Work with the function as given. Simplifying might change the domain by canceling factors that could hide restrictions.

Q: What if both numerator and denominator are zero at the same point?

This creates a removable discontinuity (a hole). The value is still excluded from the domain, but the function might be simplified by canceling common factors.

Q: How do I write the complete domain?

You can write it as: x ≠ -2/3, or in interval notation: (-∞, -2/3) ∪ (-2/3, ∞), or as a set: {x ∈ ℝ | x ≠ -2/3}.

Rational Function Domain with Zero Denominators

Look at the following function:

$\frac{2x+2}{9x+6}$

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 of definition? If so, what is it?

00:03 To find the domain of definition, remember that we cannot divide by 0

00:06 Therefore, let's find what solution makes the denominator zero

00:10 Let's isolate X

00:26 Let's factor 6 into factors 3 and 2, and 9 into factors 3 and 3

00:31 Let's simplify what we can

00:35 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:

$\frac{2x+2}{9x+6}$

What is the domain of the function?

Step-by-step solution

To solve this problem, we will determine the domain of the rational function by following these steps:

Step 1: Identify the denominator of the function, which is $9x + 6$ .
Step 2: Set the denominator equal to zero to find values of $x$ that need to be excluded from the domain: $9x + 6 = 0$ .
Step 3: Solve the equation $9x + 6 = 0$ for $x$ .
Step 4: To solve, subtract 6 from both sides to get $9x = -6$ .
Step 5: Divide each side by 9 to solve for $x$ , resulting in $x = -\frac{2}{3}$ .
Step 6: The domain of the function excludes the value $x = -\frac{2}{3}$ since it makes the denominator zero.

Thus, the domain of the given function is all real numbers except $x = -\frac{2}{3}$ , expressed as $x \ne -\frac{2}{3}$ .

Therefore, the correct choice for the domain is: $x\ne-\frac{2}{3}$ .

Final Answer

$x\ne-\frac{2}{3}$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Functions undefined when denominators equal zero
Technique: Set 9x + 6 = 0, solve to get x = -2/3
Check: Substitute x = -2/3: 9(-2/3) + 6 = -6 + 6 = 0 ✓

Common Mistakes

Avoid these frequent errors

Setting the numerator equal to zero instead of denominator
Don't set 2x + 2 = 0 to find domain restrictions = wrong excluded values! The numerator being zero makes the function equal zero, not undefined. Always set the denominator equal to zero to find values that must be excluded from the domain.

Practice Quiz

Test your knowledge with interactive questions

$22(\frac{2}{x}-1)=30$

What is the domain of the equation above?

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 or hole in the graph.

What does x ≠ -2/3 actually mean?

This notation means x can be any real number except -2/3. So x could be -1, 0, 5, or even -0.66, but never exactly -2/3.

Do I need to simplify the function first?

Not for finding the domain! Work with the function as given. Simplifying might change the domain by canceling factors that could hide restrictions.

What if both numerator and denominator are zero at the same point?

This creates a removable discontinuity (a hole). The value is still excluded from the domain, but the function might be simplified by canceling common factors.

How do I write the complete domain?

You can write it as: x ≠ -2/3, or in interval notation: (-∞, -2/3) ∪ (-2/3, ∞), or as a set: {x ∈ ℝ | x ≠ -2/3}.

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