Find the Domain of 65/(2x-2)²: Analyzing Undefined Points

Q: How do I write the final domain answer?

You can write it as "all real numbers except x ≠ 1" or in interval notation as $ (-\infty, 1) \cup (1, \infty) $.

Q: Can a rational function have multiple domain restrictions?

Yes! If you had something like $ \frac{65}{(2x-2)(x+3)} $, you'd exclude both x = 1 and x = -3 from the domain.

Rational Function Domain with Squared Denominator

Given the following function:

$\frac{65}{(2x-2)^2}$

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?

00:04 To find the domain, remember that division by 0 is not allowed

00:07 Therefore, let's find the solution that makes the denominator zero

00:16 We'll take the root to eliminate the denominator

00:26 Let's isolate X

00:41 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:

$\frac{65}{(2x-2)^2}$

What is the domain of the function?

Step-by-step solution

To solve for the domain of the function $\frac{65}{(2x-2)^2}$ , follow these steps:

Step 1: Identify when the denominator $(2x-2)^2$ equals zero because the denominator cannot be zero.
Step 2: Solve the equation $(2x-2)^2 = 0$ . This simplifies to $2x-2 = 0$ .
Step 3: Solve for $x$ by adding 2 to both sides: $2x = 2$ .
Step 4: Divide both sides by 2 to isolate $x$ : $x = 1$ .
Step 5: The value $x = 1$ makes the denominator zero, so it must be excluded from the domain.

Thus, the domain of the function is all real numbers except $x \ne 1$ .

Therefore, the solution to the problem is $x \ne 1$ .

Final Answer

$x\ne1$

Key Points to Remember

Essential concepts to master this topic

Rule: Domain excludes values that make the denominator equal zero
Technique: Set (2x-2)² = 0, solve 2x-2 = 0 to get x = 1
Check: Substitute x = 1: (2(1)-2)² = 0², making the fraction undefined ✓

Common Mistakes

Avoid these frequent errors

Forgetting to solve the equation inside the squared term
Don't just look at the squared denominator and think it can't be zero = missing the restriction! Squared expressions can equal zero when the base expression equals zero. Always set the base expression (2x-2) equal to zero and solve for x.

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 does the square in the denominator matter?

The square doesn't change when the denominator becomes zero! When $2x-2 = 0$ , then $(2x-2)^2 = 0^2 = 0$ , making the fraction undefined.

What if the denominator was cubed or raised to a higher power?

The same rule applies! Any power of zero equals zero, so you still set the base expression equal to zero and solve. The power doesn't matter for finding domain restrictions.

How do I write the final domain answer?

You can write it as "all real numbers except x ≠ 1" or in interval notation as $(-\infty, 1) \cup (1, \infty)$ .

What's the difference between this and a linear denominator?

There's no difference in finding the restriction! Whether it's $(2x-2)$ or $(2x-2)^2$ , you still solve $2x-2 = 0$ to find x = 1.

Can a rational function have multiple domain restrictions?

Yes! If you had something like $\frac{65}{(2x-2)(x+3)}$ , you'd exclude both x = 1 and x = -3 from the domain.

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