Find Where f(x) < 0: Analyzing Function with Critical Points at -10, -6, and -2

Q: How do I know where the function is negative just from looking at the graph?

Look for parts of the curve that are below the x-axis! When the graph dips below the horizontal line y = 0, those x-values make f(x) negative.

Q: What's the difference between f(x) < 0 and f(x) ≤ 0?

The symbol strictly less than, so we exclude points where f(x) = 0. The symbol ≤ includes points where f(x) equals zero. Since we want f(x)

Q: Why don't we include x = -6 in our answer?

Actually, x = -6 is included! The answer \( -10 means all values between -10 and -2, including -6, -8, -4, etc. The graph shows f(-6) is negative.

Q: How can I check if my interval is correct?

Pick a test point inside your interval and verify the function is negative there. For example, at x = -6, the graph shows the function value is below zero, confirming our answer.

Q: What if the function crosses the x-axis multiple times?

Each time the function crosses the x-axis, it changes from positive to negative (or vice versa). You need to check each interval between crossings separately to see where f(x)

Function Analysis with Critical Point Intervals

Find all values of $x$

where $f\left(x\right) < 0$ .

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Find all values of $x$

where $f\left(x\right) < 0$ .

Step-by-step solution

To solve the problem of finding all $x$ values where $f(x) < 0$ , we analyze the graph provided:

The graph of the function $f(x)$ shows it is below the x-axis in the interval from $x = -10$ to $x = -2$ . Between these points, $f(x)$ is negative because the complete span of the graph resides beneath the x-axis between these points.

Steps to validate this are:

Recognize the x-intercepts, which occur at $x = -10$ and $x = -2$ , where the curve crosses the x-axis.
The graph stays below the x-axis between these intercepts, indicating the function is negative.

Thus, the correct interval where $f(x) < 0$ is $-10 < x < -2$ .

Therefore, the solution to the problem is $-10 < x < -2$ .

Final Answer

$-10 < x < -2$

Key Points to Remember

Essential concepts to master this topic

Rule: Function is negative when graph lies below the x-axis
Technique: Identify zeros at x = -10 and x = -2, then check intervals
Check: Test point x = -6: function value is negative, confirming -10 < x < -2 ✓

Common Mistakes

Avoid these frequent errors

Including the zeros in the solution interval
Don't write $-10 \leq x \leq -2$ when asked for f(x) < 0! At x = -10 and x = -2, the function equals zero, not less than zero. Always use strict inequalities: $-10 < x < -2$ excludes the zeros where f(x) = 0.

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

How do I know where the function is negative just from looking at the graph?

Look for parts of the curve that are below the x-axis! When the graph dips below the horizontal line y = 0, those x-values make f(x) negative.

What's the difference between f(x) < 0 and f(x) ≤ 0?

The symbol < means strictly less than, so we exclude points where f(x) = 0. The symbol ≤ includes points where f(x) equals zero. Since we want f(x) < 0, we don't include the zeros at x = -10 and x = -2.

Why don't we include x = -6 in our answer?

Actually, x = -6 is included! The answer $-10 < x < -2$ means all values between -10 and -2, including -6, -8, -4, etc. The graph shows f(-6) is negative.

How can I check if my interval is correct?

Pick a test point inside your interval and verify the function is negative there. For example, at x = -6, the graph shows the function value is below zero, confirming our answer.

What if the function crosses the x-axis multiple times?

Each time the function crosses the x-axis, it changes from positive to negative (or vice versa). You need to check each interval between crossings separately to see where f(x) < 0.

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