Finding the Domain of (2x+2)/√(x+2.5): Rational Function Analysis

Q: Why can't x equal -2.5 if the square root of 0 exists?

Great question! While $ \sqrt{0} = 0 $ is defined, having 0 in the denominator makes the fraction undefined. Division by zero is never allowed in mathematics!

Q: How is this different from just having a square root in the numerator?

If $ \sqrt{x+2.5} $ were in the numerator, we'd only need $ x \geq -2.5 $. But in the denominator, we need the strict inequality $ x > -2.5 $ to avoid division by zero.

Q: What happens if I pick x = -3 for this function?

Let's see: $ x + 2.5 = -3 + 2.5 = -0.5 $. Since we can't take the square root of a negative number (in real numbers), x = -3 is not in the domain.

Q: How do I remember when to use > versus ≥?

Easy trick: If the expression with the variable is in a denominator, use strict inequality (> or ). If it's only under a square root sign, use inclusive inequality (≥ or ≤).

Q: Can I graph this to check my domain?

Absolutely! Graph the function and look where it exists. You'll see it starts just to the right of x = -2.5 and continues forever to the right. The vertical line x = -2.5 is a vertical asymptote.

Rational Function Domains with Square Root Denominators

Look at the following function:

$\frac{2x+2}{\sqrt{x+2.5}}$

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?

00:04 A root must be for a positive number greater than 0

00:10 Let's isolate X

00:17 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:

$\frac{2x+2}{\sqrt{x+2.5}}$

What is the domain of the function?

Step-by-step solution

To find the domain of the function $\frac{2x+2}{\sqrt{x+2.5}}$ , we need to determine for which values of $x$ the expression is defined.

Step 1: Identify the restriction on the square root.
The square root function $\sqrt{x+2.5}$ is defined when the expression inside the square root is non-negative. Thus, we have the inequality:

$x + 2.5 \geq 0$

Step 2: Solve the inequality for $x$ .
Subtract 2.5 from both sides:

$x \geq -2.5$

Step 3: Ensure the denominator is not zero because division by zero is undefined.
Since $x + 2.5 \neq 0$ , we require:

$x \neq -2.5$

Therefore, combining these results, the domain of the function is:

$x > -2.5$

The correct answer to the problem, represented as a choice, is:

$x > -2.5$

Final Answer

$x > -\text{2}.5$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Expression under square root must be non-negative
Technique: Set x + 2.5 > 0 to avoid zero denominator
Check: Test x = -2 in original function: defined since -2 > -2.5 ✓

Common Mistakes

Avoid these frequent errors

Using ≥ instead of > for square root in denominator
Don't write x ≥ -2.5 when square root is in denominator = includes x = -2.5 where function is undefined! Division by zero occurs when x + 2.5 = 0. Always use strict inequality x > -2.5 for denominators.

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 x equal -2.5 if the square root of 0 exists?

Great question! While $\sqrt{0} = 0$ is defined, having 0 in the denominator makes the fraction undefined. Division by zero is never allowed in mathematics!

How is this different from just having a square root in the numerator?

If $\sqrt{x+2.5}$ were in the numerator, we'd only need $x \geq -2.5$ . But in the denominator, we need the strict inequality $x > -2.5$ to avoid division by zero.

What happens if I pick x = -3 for this function?

Let's see: $x + 2.5 = -3 + 2.5 = -0.5$ . Since we can't take the square root of a negative number (in real numbers), x = -3 is not in the domain.

How do I remember when to use > versus ≥?

Easy trick: If the expression with the variable is in a denominator, use strict inequality (> or <). If it's only under a square root sign, use inclusive inequality (≥ or ≤).

Can I graph this to check my domain?

Absolutely! Graph the function and look where it exists. You'll see it starts just to the right of x = -2.5 and continues forever to the right. The vertical line x = -2.5 is a vertical asymptote.

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