Finding Negative Domains: Analyzing Function Values from -0.5 to 2

Q: What's the difference between a negative function and a decreasing function?

A negative function has y-values below the x-axis (y decreasing function slopes downward but can still have positive y-values. Don't confuse the two concepts!

Q: How do I read where the function is negative from the graph?

Look for portions of the curve that are below the horizontal x-axis. In this graph, the function dips below the x-axis when $ x > 1 $.

Q: Why isn't the answer 'x < 0' since the left side looks lower?

The left side of the graph is above the x-axis, so those y-values are still positive! Only look at where the curve actually goes below the horizontal line at y = 0.

Q: What if the graph touches the x-axis?

If the graph touches the x-axis, that point has y = 0, which is neither positive nor negative. We only want intervals where y is strictly less than zero.

Q: How can I double-check my answer?

Pick a test point in your interval! For example, if you think $ x > 1 $ is correct, check x = 1.5 on the graph. You'll see the y-value is indeed below the x-axis.

Function Analysis with Negative Value Intervals

Determine in which domain the function is negative?

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find the negative domain of the function

00:04 A function is negative when its Y values are less than 0

00:08 That is, below the X axis

00:13 At this point the function goes below the X-axis

00:18 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

Determine in which domain the function is negative?

Step-by-step solution

Remember that a function is increasing if both X values and Y values are increasing simultaneously.

A function is decreasing if X values are increasing while Y values are decreasing simultaneously.

In the graph, we can observe that in the domain $x > 1$ the function is decreasing, meaning the Y values are decreasing.

Final Answer

$x > 1$

Key Points to Remember

Essential concepts to master this topic

Negative Function Rule: Function is negative when graph lies below x-axis
Reading Technique: Follow curve from left to right, identify where y < 0
Verification: Check specific points: when x = 1.5, y is clearly below zero ✓

Common Mistakes

Avoid these frequent errors

Confusing decreasing function with negative function values
Don't assume a decreasing function means negative values = wrong domain identification! A function can decrease while still being positive. Always check where the graph sits relative to the x-axis, not just its slope direction.

Practice Quiz

Test your knowledge with interactive questions

Does the function in the graph decrease throughout?

FAQ

Everything you need to know about this question

What's the difference between a negative function and a decreasing function?

A negative function has y-values below the x-axis (y < 0). A decreasing function slopes downward but can still have positive y-values. Don't confuse the two concepts!

How do I read where the function is negative from the graph?

Look for portions of the curve that are below the horizontal x-axis. In this graph, the function dips below the x-axis when $x > 1$ .

Why isn't the answer 'x < 0' since the left side looks lower?

The left side of the graph is above the x-axis, so those y-values are still positive! Only look at where the curve actually goes below the horizontal line at y = 0.

What if the graph touches the x-axis?

If the graph touches the x-axis, that point has y = 0, which is neither positive nor negative. We only want intervals where y is strictly less than zero.

How can I double-check my answer?

Pick a test point in your interval! For example, if you think $x > 1$ is correct, check x = 1.5 on the graph. You'll see the y-value is indeed below the x-axis.

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