Find the Domain of (5+4x)/(x-3)²: Rational Function Analysis

Q: Why does the square in the denominator matter for the domain?

Even though $ (x-3)^2 $ is always positive or zero, it still equals zero when x = 3. Division by zero is undefined, so we must exclude x = 3 from the domain.

Q: Does the numerator 5+4x affect the domain?

No! The numerator can equal zero without affecting the domain. Only when the denominator equals zero do we have problems, since division by zero is undefined.

Q: How do I write the domain correctly?

Write it as all real numbers except x = 3, or in notation: $ x \in \mathbb{R}, x \neq 3 $, or in interval notation: $ (-\infty, 3) \cup (3, \infty) $.

Q: Why can't x equal 3 in this function?

When x = 3, the denominator becomes $ (3-3)^2 = 0^2 = 0 $. This makes the fraction $ \frac{17}{0} $, which is undefined in mathematics.

Rational Function Domains with Perfect Squares

Given the following function:

$\frac{5+4x}{(x-3)^2}$

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 we cannot divide by 0

00:06 So let's see what solution zeroes the denominator

00:09 We'll take the root to get rid of the exponent

00:20 Let's isolate X

00:27 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:

$\frac{5+4x}{(x-3)^2}$

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

Step-by-step solution

When we have an unknown in a fraction, we immediately know that the domain must exclude where the denominator equals 0.

Therefore, we need to check in which case the denominator can become 0.

Let's set:

(x-3)²≠0

Let's take the square root of both sides:

x-3≠0

Let's isolate x:

x≠3

And this is the solution we were looking for, X cannot be 3, because then it wouldn't satisfy the domain we found.

Final Answer

Yes, $x\ne3$

Key Points to Remember

Essential concepts to master this topic

Rule: Domain excludes all values where denominator equals zero
Technique: Set (x-3)² ≠ 0, solve x-3 ≠ 0 to get x ≠ 3
Check: Substitute x = 3: (3-3)² = 0 makes function undefined ✓

Common Mistakes

Avoid these frequent errors

Forgetting that squared terms still create restrictions
Don't think (x-3)² = 0 has no solution because it's squared = wrong domain! Even perfect squares equal zero when the base equals zero. Always solve the base expression: x-3 = 0 gives x = 3 to exclude.

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 does the square in the denominator matter for the domain?

Even though $(x-3)^2$ is always positive or zero, it still equals zero when x = 3. Division by zero is undefined, so we must exclude x = 3 from the domain.

Does the numerator 5+4x affect the domain?

No! The numerator can equal zero without affecting the domain. Only when the denominator equals zero do we have problems, since division by zero is undefined.

How do I write the domain correctly?

Write it as all real numbers except x = 3, or in notation: $x \in \mathbb{R}, x \neq 3$ , or in interval notation: $(-\infty, 3) \cup (3, \infty)$ .

What if the denominator had multiple factors?

Set each factor ≠ 0 separately! For example, if denominator is $(x-2)(x+1)$ , then exclude both x = 2 and x = -1 from the domain.

Why can't x equal 3 in this function?

When x = 3, the denominator becomes $(3-3)^2 = 0^2 = 0$ . This makes the fraction $\frac{17}{0}$ , which is undefined in mathematics.

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