Find all values of x where $ f(x) < 0 $.XXXYYY-4-4-4

The function is negative for all values of $ x $.

The function has no negative domain.

Determine X-Values Where f(x) is Less Than Zero: A Graphical Analysis

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

f(x) = 0 means the graph touches or crosses the x-axis (at x = -4 here). f(x) means the graph is below the x-axis. They're different conditions!

Q: How do I know if the graph goes below the x-axis on both sides?

Look carefully at the graph! You can see it dips below the x-axis both to the left and right of x = -4. The parabola opens downward and only touches the axis at one point.

Q: Why isn't x = -4 included in the solution?

At x = -4, the function value is exactly zero, not negative. We need \( f(x) , which means strictly less than zero.

Q: Can I write the answer as one inequality instead of two?

No! Since x = -4 is excluded, you need two separate inequalities: $ x OR \( x > -4 $. This covers all real numbers except -4.

Q: What if I can't see the graph clearly?

Focus on key features: find where the graph crosses or touches the x-axis (roots), then determine which intervals make the function positive or negative by checking sample points.

Graphical Analysis with Function Negativity

Find all values of x

where $f(x) < 0$ .

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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 $f(x) < 0$ .

Step-by-step solution

Let's analyze the graph to determine where $f(x) < 0$ .

The process to follow is:

Identify the x-axis intersections (roots) where $f(x) = 0$ .
Notice where the graph dips below the x-axis, indicating $f(x) < 0$ .
The graph crosses and only touches the x-axis at $x = -4$ .
The graph lies below the x-axis both to the left and right of $x = -4$ .

From this analysis, the function $f(x)$ is negative for all $x$ except at $x = -4$ , where it touches but doesn’t dip below the x-axis.

Therefore, the solution is that the function is negative for $x > -4$ or $x < -4$ .

Final Answer

$x > -4$ or $x < -4$

Key Points to Remember

Essential concepts to master this topic

Identification: Look for where the graph lies below the x-axis
Technique: Find roots first, then check intervals: at x = -4 the graph touches axis
Check: Verify graph is below x-axis for all x except x = -4 ✓

Common Mistakes

Avoid these frequent errors

Confusing where function touches axis with where it's negative
Don't say f(x) < 0 at x = -4 just because the graph touches there = wrong answer! At x = -4, f(x) = 0, not negative. Always identify where the graph is actually below the x-axis, not just touching it.

Practice Quiz

Test your knowledge with interactive questions

The graph of the function below does not intersect the $$ x $$ -axis.

The parabola's vertex is marked A.

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 graph touches or crosses the x-axis (at x = -4 here). f(x) < 0 means the graph is below the x-axis. They're different conditions!

How do I know if the graph goes below the x-axis on both sides?

Look carefully at the graph! You can see it dips below the x-axis both to the left and right of x = -4. The parabola opens downward and only touches the axis at one point.

Why isn't x = -4 included in the solution?

At x = -4, the function value is exactly zero, not negative. We need $f(x) < 0$ , which means strictly less than zero.

Can I write the answer as one inequality instead of two?

No! Since x = -4 is excluded, you need two separate inequalities: $x < -4$ OR $x > -4$ . This covers all real numbers except -4.

What if I can't see the graph clearly?

Focus on key features: find where the graph crosses or touches the x-axis (roots), then determine which intervals make the function positive or negative by checking sample points.

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