Domain Analysis of (8+8x)/(x+6)²: Finding Valid Input Values

Q: Why can't x equal -6 in this function?

When $ x = -6 $, the denominator $ (x+6)^2 $ becomes $ (-6+6)^2 = 0^2 = 0 $. Since we can't divide by zero, the function is undefined at this point.

Q: Does the numerator affect the domain?

No! The numerator $ 8+8x $ can equal zero without affecting the domain. Only when the denominator equals zero does the function become undefined.

Q: What does the squared term (x+6)² mean for the domain?

The square doesn't change which values are excluded! Since $ (x+6)^2 = 0 $ only when $ x+6 = 0 $, we still exclude $ x = -6 $. The square just means this zero has multiplicity 2.

Q: How do I write the domain in interval notation?

The domain is all real numbers except $ x = -6 $, written as: $ (-\infty, -6) \cup (-6, \infty) $. The union symbol connects the two intervals.

Q: Can a function have no domain restrictions?

Yes! Polynomial functions like $ f(x) = x^2 + 3x - 1 $ have no denominators, so their domain is all real numbers with no restrictions.

Rational Function Domains with Squared Denominators

Given the following function:

$\frac{8+8x}{(x+6)^2}$

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 see what solution makes the denominator zero

00:11 We'll take the root to eliminate the exponent

00:18 Let's isolate X

00:26 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:

$\frac{8+8x}{(x+6)^2}$

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

Step-by-step solution

To determine the domain of the function $\frac{8+8x}{(x+6)^2}$ , we must find the values of $x$ that make the function undefined.

This function is a rational function with the numerator $8+8x$ and the denominator $(x+6)^2$ . A rational function is undefined where its denominator is equal to zero.

Therefore, we need to solve for $x$ where the denominator equals zero:

Set the denominator equal to zero: $(x+6)^2 = 0$ .
Take the square root of both sides, resulting in $x+6 = 0$ .
Solve for $x$ by subtracting 6 from both sides: $x = -6$ .

This calculation shows that the function is undefined when $x = -6$ . Therefore, the domain of the function includes all real numbers except $x = -6$ .

Thus, the domain of the function is all real numbers except $x \ne -6$ .

The correct choice is:

Yes, $x\ne-6$

Final Answer

Yes, $x\ne-6$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Rational functions undefined when denominator equals zero
Technique: Set $(x+6)^2 = 0$ gives $x = -6$
Check: Substitute $x = -6$ makes denominator zero, confirming exclusion ✓

Common Mistakes

Avoid these frequent errors

Forgetting to exclude values that make denominator zero
Don't assume all real numbers work without checking the denominator = function becomes undefined! Division by zero is impossible in mathematics. Always find where the denominator equals zero and exclude those x-values from the domain.

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 x equal -6 in this function?

When $x = -6$ , the denominator $(x+6)^2$ becomes $(-6+6)^2 = 0^2 = 0$ . Since we can't divide by zero, the function is undefined at this point.

Does the numerator affect the domain?

No! The numerator $8+8x$ can equal zero without affecting the domain. Only when the denominator equals zero does the function become undefined.

What does the squared term (x+6)² mean for the domain?

The square doesn't change which values are excluded! Since $(x+6)^2 = 0$ only when $x+6 = 0$ , we still exclude $x = -6$ . The square just means this zero has multiplicity 2.

How do I write the domain in interval notation?

The domain is all real numbers except $x = -6$ , written as: $(-\infty, -6) \cup (-6, \infty)$ . The union symbol connects the two intervals.

What if there were multiple restrictions?

If the denominator had multiple factors that could equal zero, you'd exclude all those x-values. For example, $\frac{1}{(x-2)(x+3)}$ would exclude both $x = 2$ and $x = -3$ .

Can a function have no domain restrictions?

Yes! Polynomial functions like $f(x) = x^2 + 3x - 1$ have no denominators, so their domain is all real numbers with no restrictions.

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Domain Analysis of (8+8x)/(x+6)²: Finding Valid Input Values

Rational Function Domains with Squared Denominators

❤️ Continue Your Math Journey!

Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why can't x equal -6 in this function?

Does the numerator affect the domain?

What does the squared term (x+6)² mean for the domain?

How do I write the domain in interval notation?

What if there were multiple restrictions?

Can a function have no domain restrictions?

🌟 Unlock Your Math Potential

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