Domain of 65/(2x-2)²: Find Valid Input Values

Q: What if the denominator had multiple terms?

Set the entire denominator equal to zero and solve! For example, if you had $ \frac{65}{x^2-1} $, solve $ x^2-1=0 $ to get x = ±1.

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

You can write it as: All real numbers except x = 1, or in interval notation: $ (-\infty, 1) \cup (1, \infty) $, or as a restriction: $ x \neq 1 $.

Q: What happens if I try to plug in x = 1?

You get $ \frac{65}{0} $, which is undefined! Division by zero has no meaning in mathematics, which is why x = 1 must be excluded from the domain.

Q: Can a rational function have more than one domain restriction?

Absolutely! If the denominator factors into multiple terms, each factor that equals zero gives a restriction. For example, $ \frac{1}{(x-2)(x+3)} $ has restrictions at both x = 2 and x = -3.

Domain Restrictions with Rational Functions

Given the following function:

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

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

00:10 First, let's see if the function has a domain. If yes, what would it be?

00:15 To find the domain, remember: division by zero is not allowed.

00:20 So, let's find what makes the denominator equal to zero.

00:25 We can take the square root to remove the exponent.

00:29 Now, let's isolate the variable X.

00:42 And there we have it! That's the solution to our question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Given the following function:

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

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

Step-by-step solution

The denominator of the function cannot be equal to 0.

Therefore, we will set the denominator equal to 0 and solve for the domain:

$(2x-2)^2\ne0$

$2x\ne2$

$x\ne1$

In other words, the domain of the function is all numbers except 1.

Final Answer

Yes, $x\ne1$

Key Points to Remember

Essential concepts to master this topic

Rule: Rational functions are undefined when denominators equal zero
Technique: Set (2x-2)² = 0, solve 2x = 2, get x = 1
Check: Substitute x = 1: (2(1)-2)² = 0² = 0 in denominator ✓

Common Mistakes

Avoid these frequent errors

Ignoring the squared term in the denominator
Don't solve 2x - 2 = 0 while ignoring the square = finding wrong restrictions! The square doesn't change which values make the expression zero, but students often get confused. Always solve the expression inside parentheses first, then consider the exponent.

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 (2x-2)² matter for finding domain?

The square doesn't change which x-values make the denominator zero! Since $(2x-2)^2 = 0$ only when $2x-2 = 0$ , we still get x = 1 as the restriction.

What if the denominator had multiple terms?

Set the entire denominator equal to zero and solve! For example, if you had $\frac{65}{x^2-1}$ , solve $x^2-1=0$ to get x = ±1.

How do I write the domain in proper notation?

You can write it as: All real numbers except x = 1, or in interval notation: $(-\infty, 1) \cup (1, \infty)$ , or as a restriction: $x \neq 1$ .

What happens if I try to plug in x = 1?

You get $\frac{65}{0}$ , which is undefined! Division by zero has no meaning in mathematics, which is why x = 1 must be excluded from the domain.

Can a rational function have more than one domain restriction?

Absolutely! If the denominator factors into multiple terms, each factor that equals zero gives a restriction. For example, $\frac{1}{(x-2)(x+3)}$ has restrictions at both x = 2 and x = -3.

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