Find the Domain for the Function: (2x+2)/(3x-1)

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, so we must exclude it from the domain.

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

The domain is $ (-\infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty) $. This shows all real numbers except $ x = \frac{1}{3} $.

Q: What if there are multiple fractions in the function?

Find where each denominator equals zero separately, then exclude all those values from the domain. The domain restrictions are the union of all problematic values.

Q: Can the domain ever be all real numbers for a rational function?

Only if the denominator is a constant (like 5 or -2). If the denominator contains a variable, there will always be at least one value to exclude from the domain.

Domain Finding with Rational Functions

Look at the following function:

$\frac{2x+2}{3x-1}$

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? If so, what is it?

00:03 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

00:10 Let's isolate X

00:19 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:

$\frac{2x+2}{3x-1}$

What is the domain of the function?

Step-by-step solution

To find the domain of the function $\frac{2x + 2}{3x - 1}$ , we must ensure that the function is defined for all values of $x$ except where the denominator is zero.

Follow these steps to determine the domain:

Step 1: Identify the denominator of the function. The denominator is $3x - 1$ .
Step 2: Set the denominator equal to zero and solve for $x$ : $3x - 1 = 0$
Step 3: Solve the equation $3x - 1 = 0$ : $3x = 1$ $x = \frac{1}{3}$
Step 4: State the domain. Since the function is undefined at $x = \frac{1}{3}$ , the domain consists of all real numbers except $x = \frac{1}{3}$ .

The domain of the function $\frac{2x + 2}{3x - 1}$ is therefore all real numbers $x$ such that $x \neq \frac{1}{3}$ .

Thus, the solution is: $x \ne \frac{1}{3}$ .

Final Answer

$x\ne\frac{1}{3}$

Key Points to Remember

Essential concepts to master this topic

Rule: Domain excludes values that make the denominator zero
Technique: Set 3x - 1 = 0, solve to get x = 1/3
Check: Substitute x = 1/3: denominator becomes 3(1/3) - 1 = 0 ✓

Common Mistakes

Avoid these frequent errors

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

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, so we must exclude it from the domain.

What if the numerator is also zero when x = 1/3?

Even if both numerator and denominator are zero, we still exclude that value from the domain. The expression $\frac{0}{0}$ is indeterminate, not defined.

How do I write the domain in interval notation?

The domain is $(-\infty, \frac{1}{3}) \cup (\frac{1}{3}, \infty)$ . This shows all real numbers except $x = \frac{1}{3}$ .

What if there are multiple fractions in the function?

Find where each denominator equals zero separately, then exclude all those values from the domain. The domain restrictions are the union of all problematic values.

Can the domain ever be all real numbers for a rational function?

Only if the denominator is a constant (like 5 or -2). If the denominator contains a variable, there will always be at least one value to exclude from the domain.

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