Finding the Domain of the Function 9x/4: Complete Analysis

Q: Why isn't x = 0 excluded from the domain?

Because x = 0 is in the numerator, not the denominator. When x = 0, we get $ \frac{9(0)}{4} = \frac{0}{4} = 0 $, which is perfectly valid!

Q: What makes a function have domain restrictions?

Domain restrictions occur when operations become undefined, like dividing by zero, taking square roots of negatives, or logarithms of non-positive numbers.

Q: How do I know if the denominator will ever equal zero?

Set the denominator equal to zero and solve. For $ \frac{9x}{4} $, the denominator is just 4, which never equals zero, so no restrictions!

Q: Can linear functions ever have domain restrictions?

Simple linear functions like $ y = mx + b $ have no restrictions. But rational functions like $ y = \frac{mx + b}{cx + d} $ can have restrictions when the denominator equals zero.

Domain Analysis with Linear Functions

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

$\frac{9x}{4}$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Does the function have a domain, and if so, what is it?

00:10 To find the domain, remember: division by zero is not allowed.

00:15 The denominator here is a non-zero constant, so there are no domain restrictions.

00:21 We have a domain restriction if the denominator contains a variable.

00:26 And that's how we solve this question!

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

$\frac{9x}{4}$

Step-by-step solution

Since the function's denominator equals 4, the domain of the function is all real numbers. This means that any one of the x values could be compatible.

Final Answer

No, the entire domain

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Check denominators for values that make them zero
Technique: Since denominator is constant 4, no restrictions apply
Check: Test any x-value: $\frac{9(0)}{4} = 0$ works ✓

Common Mistakes

Avoid these frequent errors

Confusing numerator restrictions with denominator restrictions
Don't think x = 0 is restricted because it's in the numerator = wrong domain! The numerator can be zero without problems. Always check only if the denominator equals zero for any x-value.

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 isn't x = 0 excluded from the domain?

Because x = 0 is in the numerator, not the denominator. When x = 0, we get $\frac{9(0)}{4} = \frac{0}{4} = 0$ , which is perfectly valid!

What makes a function have domain restrictions?

Domain restrictions occur when operations become undefined, like dividing by zero, taking square roots of negatives, or logarithms of non-positive numbers.

How do I know if the denominator will ever equal zero?

Set the denominator equal to zero and solve. For $\frac{9x}{4}$ , the denominator is just 4, which never equals zero, so no restrictions!

What would the domain be if the function was 9x/(x-4)?

Then you'd have a restriction! The denominator x - 4 = 0 when x = 4, so the domain would be all real numbers except x ≠ 4.

Can linear functions ever have domain restrictions?

Simple linear functions like $y = mx + b$ have no restrictions. But rational functions like $y = \frac{mx + b}{cx + d}$ can have restrictions when the denominator equals zero.

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