Finding X Values Where f(x) > 0: Graph Analysis Solution

Q: What's the difference between f(x) > 0 and f(x) ≥ 0?

f(x) > 0 means the function is strictly positive (above the x-axis), while f(x) ≥ 0 includes points where the function equals zero (touching the x-axis). In this problem, we need strictly positive values only.

Q: How do I know if the graph goes above the x-axis?

Look at the vertical position of the curve relative to the horizontal x-axis. If any part of the curve is in the upper half of the coordinate plane, then $ f(x) > 0 $ for those x-values.

Q: Why isn't x = 8 part of the answer?

At x = 8, the function touches the x-axis, meaning $ f(8) = 0 $. Since we need $ f(x) > 0 $ (strictly greater than zero), points where the function equals zero don't count.

Q: What if I can't see the exact shape of the graph?

Focus on key points where the graph intersects or touches the x-axis. Then trace the curve between these points to see if it goes above (positive) or below (negative) the x-axis.

Q: Could there be positive values I'm missing?

Always scan the entire visible graph from left to right. If the curve consistently stays on or below the x-axis everywhere except touching at one point, then there are no positive values.

Graph Analysis with Function Positivity

Based on the graph data, find for which X values the function graph $f\left(x\right) > 0$

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Based on the graph data, find for which X values the function graph $f\left(x\right) > 0$

Step-by-step solution

To solve this problem, we will perform a graphical analysis of where the function $f(x)$ is greater than zero:

Locate where the graph of $f(x)$ lies above the x-axis, as this indicates positive values of $f(x)$ .
Examine the graph from the bottom to the top to determine intervals or specific points either crossing or not crossing the x-axis.
Determine if there are any specific segments over which the function is above the x-axis.

By analyzing the graph provided, we observe the path of the curve of $f(x)$ and see that it consistently stays on or below the x-axis, indicating non-positive values. Particularly:

The point $x = 8$ corresponds to touching the x-axis, indicating zero value, not positive.
Other than this point, the graph doesn't traverse above the x-axis, confirming non-positive values elsewhere.

Thus, the function $f(x)$ has no intervals where it is positive.

Therefore, the correct conclusion is that the function $f(x)$ has no values where $f\left(x\right) > 0$ .

Final Answer

No such values.

Key Points to Remember

Essential concepts to master this topic

Rule: Function is positive when graph lies above x-axis
Technique: Check where curve crosses or touches x-axis at x = 8
Check: Verify graph never goes above x-axis anywhere ✓

Common Mistakes

Avoid these frequent errors

Confusing touching the x-axis with being above it
Don't assume $f(x) = 0$ means $f(x) > 0$ = wrong conclusion! When a graph only touches the x-axis at x = 8, that point has zero value, not positive. Always distinguish between $f(x) = 0$ (touching) and $f(x) > 0$ (above).

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below intersects the X-axis at points A and B.

The vertex of the parabola is marked at point C.

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

FAQ

Everything you need to know about this question

What's the difference between f(x) > 0 and f(x) ≥ 0?

f(x) > 0 means the function is strictly positive (above the x-axis), while f(x) ≥ 0 includes points where the function equals zero (touching the x-axis). In this problem, we need strictly positive values only.

How do I know if the graph goes above the x-axis?

Look at the vertical position of the curve relative to the horizontal x-axis. If any part of the curve is in the upper half of the coordinate plane, then $f(x) > 0$ for those x-values.

Why isn't x = 8 part of the answer?

At x = 8, the function touches the x-axis, meaning $f(8) = 0$ . Since we need $f(x) > 0$ (strictly greater than zero), points where the function equals zero don't count.

What if I can't see the exact shape of the graph?

Focus on key points where the graph intersects or touches the x-axis. Then trace the curve between these points to see if it goes above (positive) or below (negative) the x-axis.

Could there be positive values I'm missing?

Always scan the entire visible graph from left to right. If the curve consistently stays on or below the x-axis everywhere except touching at one point, then there are no positive values.

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