Finding the Positive Domain: Linear Function with Points (0,a) and (2a,0)

Linear Functions with Domain Analysis

Look at the function in the figure.

What is the positive domain of the function?

xy(0,a)(2a,0)

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find the domain of positivity of the function
00:03 The function is positive when it's above the X-axis
00:06 and negative when the function is below the X-axis
00:15 Let's identify when the function intersects the X-axis
00:19 We'll identify when the function is positive and when it's negative
00:22 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Look at the function in the figure.

What is the positive domain of the function?

xy(0,a)(2a,0)

2

Step-by-step solution

Positive domain is another name for the point from which the x values are positive and not negative.

From the figure, it can be seen that the function ascends and passes through the intersection point with the X-axis (where X is equal to 0) at point 2a.

Therefore, it is possible to understand that from the moment X is greater than 2a, the function is in the domains of positivity.

Therefore, the function is positive when:

2a<x 2a < x

3

Final Answer

2a<x 2a < x

Key Points to Remember

Essential concepts to master this topic
  • Definition: Positive domain means where function values are above zero
  • Technique: Find x-intercept where function crosses x-axis at (2a,0)
  • Check: Test values: when x > 2a, function is positive ✓

Common Mistakes

Avoid these frequent errors
  • Confusing positive domain with positive x-values
    Don't think positive domain means where x is positive = wrong interpretation! This confuses the input (x-values) with output (function values). Always remember positive domain means where the function output is above the x-axis.

Practice Quiz

Test your knowledge with interactive questions

Look at the linear function represented in the diagram.

When is the function positive?

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333000

FAQ

Everything you need to know about this question

What's the difference between positive domain and where x is positive?

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Positive domain refers to where the function values (y-values) are positive, not the x-values. Look at where the graph is above the x-axis, regardless of whether x itself is positive or negative.

How do I find where a linear function is positive?

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Find the x-intercept where the line crosses the x-axis (where y = 0). For an increasing line, the function is positive to the right of this point. For a decreasing line, it's positive to the left.

Why is the answer 2a < x instead of x > 2a?

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Both expressions mean the same thing! 2a<x 2a < x and x>2a x > 2a are equivalent. The first format emphasizes that x must be greater than the boundary value 2a.

What if the line was decreasing instead of increasing?

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For a decreasing line, the positive domain would be to the left of the x-intercept. Since this line has a negative slope (goes down from left to right), you'd look where x < 2a instead.

How can I tell if the line is increasing or decreasing from the graph?

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Look at the line's direction: if it goes up from left to right, it's increasing (positive slope). If it goes down from left to right, it's decreasing (negative slope).

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