Solve f(x) < 0: Finding Points Where Function is Negative

Q: How do I know which parts of the graph are below the x-axis?

Look for sections where the curve dips below the horizontal line (x-axis). In this graph, the function goes below the x-axis on both sides of x = 8.

Q: What does f(x) < 0 actually mean?

f(x) means the function's output values are negative. On a graph, this appears as the curve being below the x-axis since negative y-values are plotted downward.

Q: Why isn't x = 8 included in the answer?

Because at x = 8, we have f(8) = 0, not f(8) strictly less than zero, so we exclude points where it equals zero.

Q: How can I verify my answer without the graph?

Test points from each region! Pick a value like x = 0 (should give negative result) and x = 10 (should also give negative result) to confirm both intervals work.

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

Then you'd have multiple intervals to consider! Check each region between crossings separately. Some intervals might be above the axis (positive) and others below (negative).

Function Inequality with Graph Analysis

Find all values of $x$

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

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Understand the problem

Find all values of $x$

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

Step-by-step solution

To solve the problem of finding where $f(x) < 0$ , we need to analyze the graph of the function:

Examine the graph to identify the intervals where it lies below the $x$ -axis.
The graph crosses the $x$ -axis at two key points, $x = 8$ and the line extends indefinitely.
Thus, the function $f(x)$ is negative when it moves below the $x$ -axis on either side of $x=8$ .

From this graphical analysis, $f(x)$ is negative for:

The range $x < 8$ : The part of the graph to the left of $x = 8$ is under the $x$ -axis.
The range $x > 8$ : The part of the graph to the right of $x = 8$ is also under the $x$ -axis.

Therefore, the solution to the problem is that $f(x) < 0$ for $x < 8$ or $x > 8$ .

This matches the answer choice $x < 8$ or $8 < x$ .

Final Answer

$x < 8$ or $8 < x$

Key Points to Remember

Essential concepts to master this topic

Reading Graphs: Function is negative when curve lies below x-axis
Technique: Find x-intercepts first, then check regions: f(8) = 0
Check: Verify by testing points: pick x = 0 and x = 10 ✓

Common Mistakes

Avoid these frequent errors

Only looking at x-intercepts instead of regions
Don't just identify where f(x) = 0 and stop there = incomplete answer! The x-intercepts show where the function crosses the axis, but you need the intervals where it's below. Always examine which regions lie below the x-axis.

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 which parts of the graph are below the x-axis?

Look for sections where the curve dips below the horizontal line (x-axis). In this graph, the function goes below the x-axis on both sides of x = 8.

What does f(x) < 0 actually mean?

f(x) < 0 means the function's output values are negative. On a graph, this appears as the curve being below the x-axis since negative y-values are plotted downward.

Why isn't x = 8 included in the answer?

Because at x = 8, we have f(8) = 0, not f(8) < 0. The question asks for where the function is strictly less than zero, so we exclude points where it equals zero.

How can I verify my answer without the graph?

Test points from each region! Pick a value like x = 0 (should give negative result) and x = 10 (should also give negative result) to confirm both intervals work.

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

Then you'd have multiple intervals to consider! Check each region between crossings separately. Some intervals might be above the axis (positive) and others below (negative).

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