Why is the answer format 'x 0: x≠5'?

This notation is misleading but follows the question format. It should really say:Negative domain (where y nonePositive domain (where y > 0): all x except x = 5The 'x 0' labels refer to the sign of the function output, not the input.

Find the Domain of y=(x-5)²: Analyzing Positive and Negative Inputs

Q: What does 'positive domain' and 'negative domain' actually mean?

Positive domain: all x-values where the function output $ y > 0 $Negative domain: all x-values where the function output \( y It's about the function's output, not whether x is positive or negative!

Q: Why can't a squared expression be negative?

When you square any real number, the result is always positive or zero. For example: $ 3^2 = 9 $ and $ (-3)^2 = 9 $. Even $ 0^2 = 0 $. There's no real number that gives a negative result when squared!

Q: How do I find where the function equals zero?

Set the function equal to zero: $ (x-5)^2 = 0 $. Since only zero squared equals zero, we need $ x-5 = 0 $, so $ x = 5 $. This is the only point where the function touches the x-axis.

Q: What does 'x ≠ 5' mean in the positive domain?

This means all real numbers except 5. At every x-value except x = 5, the function $ y = (x-5)^2 $ produces a positive output. At x = 5, the output is exactly zero, so it's neither positive nor negative.

Quadratic Functions with Domain Analysis

Find the positive and negative domains of the function below:

$y=\left(x-5\right)^2$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find the positive and negative domains of the function below:

$y=\left(x-5\right)^2$

Step-by-step solution

To determine the positive and negative domains of the function $y = (x - 5)^2$ , follow these steps:

Step 1: Recognize the function is in vertex form: $y = (x - h)^2 + k$ , where the vertex is $(h, k) = (5, 0)$ .
Step 2: Observe the properties of a square. The expression $(x - 5)^2$ is always greater than or equal to zero because a square of any real number cannot be negative.
Step 3: Analyze when the function is zero. The output $y = 0$ when $x - 5 = 0$ , or $x = 5$ .
Step 4: Consider where $y$ is positive. For all $x$ except $x = 5$ , the square $(x-5)^2 > 0$ .
Step 5: Determine where $y$ is negative. Since a squared term is never negative, there are no values of $x$ for which $y < 0$ .

Based on these steps, the positive domain captures all $x$ except where $x = 5$ , and the negative domain is nonexistent because the square is always non-negative.

Therefore, the solution is:

$x < 0 :$ none
$x > 0 : x\ne5$

Final Answer

$x < 0 :$ none
$x > 0 : x\ne5$

Key Points to Remember

Essential concepts to master this topic

Domain Rule: Squared expressions are always non-negative for real numbers
Technique: Set $(x-5)^2 = 0$ to find where y equals zero
Check: Verify at x=5: $(5-5)^2 = 0$ , at x=6: $(6-5)^2 = 1 > 0$ ✓

Common Mistakes

Avoid these frequent errors

Confusing positive/negative domains with positive/negative x-values
Don't think positive domain means x > 0 = wrong interpretation! The question asks where the function OUTPUT (y-values) is positive or negative, not the input x-values. Always focus on when y > 0 or y < 0.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below does not intersect the $$ x $$ -axis.

The parabola's vertex is marked A.

Find all values of $$ x $$ where
$f\left(x\right) > 0$ .

FAQ

Everything you need to know about this question

What does 'positive domain' and 'negative domain' actually mean?

Positive domain: all x-values where the function output $y > 0$
Negative domain: all x-values where the function output $y < 0$
It's about the function's output, not whether x is positive or negative!

Why can't a squared expression be negative?

When you square any real number, the result is always positive or zero. For example: $3^2 = 9$ and $(-3)^2 = 9$ . Even $0^2 = 0$ . There's no real number that gives a negative result when squared!

How do I find where the function equals zero?

Set the function equal to zero: $(x-5)^2 = 0$ . Since only zero squared equals zero, we need $x-5 = 0$ , so $x = 5$ . This is the only point where the function touches the x-axis.

What does 'x ≠ 5' mean in the positive domain?

This means all real numbers except 5. At every x-value except x = 5, the function $y = (x-5)^2$ produces a positive output. At x = 5, the output is exactly zero, so it's neither positive nor negative.

Why is the answer format 'x < 0: none' and 'x > 0: x≠5'?

This notation is misleading but follows the question format. It should really say:
Negative domain (where y < 0): none
Positive domain (where y > 0): all x except x = 5
The 'x < 0' and 'x > 0' labels refer to the sign of the function output, not the input.

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Find the Domain of y=(x-5)²: Analyzing Positive and Negative Inputs

Quadratic Functions with Domain Analysis

❤️ Continue Your Math Journey!

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

What does 'positive domain' and 'negative domain' actually mean?

Why can't a squared expression be negative?

How do I find where the function equals zero?

What does 'x ≠ 5' mean in the positive domain?

Why is the answer format 'x < 0: none' and 'x > 0: x≠5'?

🌟 Unlock Your Math Potential

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