Function Graph Analysis: Determining Continuous Decrease Behavior

Function Analysis with Monotonic Behavior

Does the function in the graph decrease throughout?

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Step-by-step written solution

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1

Understand the problem

Does the function in the graph decrease throughout?

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2

Step-by-step solution

To solve this problem, we'll begin by examining the graph of the function provided:

  • Step 1: Observe the graph from left to right along the x-axis.
  • Step 2: Look for any intervals where the function value (y-coordinate) does not decrease as the x-value increases.
  • Step 3: Pay special attention to segments where the graph might look horizontal or rising.

Upon inspecting the graph, we find:

- There are sections where the function's y-values appear to remain constant or potentially rise as the x-values increase. Specifically, even if the function decreases in major portions, any interval where it doesn't means the function cannot be classified as decreasing throughout.

Thus, the function does not strictly decrease on the entire interval shown. Therefore, the solution to the problem is No.

3

Final Answer

No

Key Points to Remember

Essential concepts to master this topic
  • Definition: Decreasing function means y-values get smaller as x increases
  • Method: Trace graph left to right, checking if any sections rise
  • Verification: Find any horizontal or rising segments to prove not decreasing throughout ✓

Common Mistakes

Avoid these frequent errors
  • Focusing only on steep declining sections
    Don't just look at the obvious downward slopes and ignore flat or slightly rising parts = wrong conclusion! Even tiny horizontal segments mean the function isn't decreasing throughout. Always examine the entire graph systematically from left to right.

Practice Quiz

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Does the function in the graph decrease throughout?

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FAQ

Everything you need to know about this question

What exactly does 'decreasing throughout' mean?

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A function decreases throughout when every single point has a smaller y-value than all points to its left. If even one tiny section stays flat or goes up, the function is not decreasing throughout.

How can I tell if a section is truly flat or just looks flat?

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Look carefully at the graph's path. Even if a section appears nearly horizontal, if it's not perfectly flat or if it rises even slightly, the function isn't decreasing there.

Does the function need to decrease at the same rate everywhere?

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No! A decreasing function can decrease slowly in some parts and quickly in others. What matters is that it never stops decreasing - no flat spots, no increases allowed.

What if I can't see the exact behavior in a small section?

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When in doubt, look for any evidence of non-decreasing behavior. Even one questionable flat section is enough to conclude the function doesn't decrease throughout.

Can a function be decreasing if it has breaks or jumps?

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Yes, but you analyze each continuous piece separately. However, this graph appears to be one continuous curve, so examine the whole path for any non-decreasing segments.

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