Find Where f(x) < 0: Analyzing Function Values Between x = 1 and x = 7

Q: How do I know when f(x) is negative just by looking at the graph?

Look for where the curve is below the x-axis. If you can draw a horizontal line from any point on the curve down to the x-axis, then f(x)

Q: What do the x-intercepts tell me about the inequality?

X-intercepts are where f(x) = 0, which act as boundary points. The function changes sign (from positive to negative or vice versa) at these points.

Q: Why is the answer 1 < x < 7 and not 1 ≤ x ≤ 7?

We need f(x) (strictly less than zero). At x = 1 and x = 7, the function equals zero, not less than zero, so we use

Q: What if the graph touches but doesn't cross the x-axis?

If the graph just touches the x-axis at a point, that's still an x-intercept where f(x) = 0. The function doesn't change sign there, so check the intervals carefully.

Q: How can I double-check my answer?

Pick a test point between your boundaries, like x = 4. Look at the graph: is f(4) above or below the x-axis? If below, then f(4)

Function Inequalities with Graph Analysis

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

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Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

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

Step-by-step solution

To solve the problem of determining where $f(x) < 0$ :

Step 1: Identify the x-intercepts of the graph, which are $x = 1$ and $x = 7$ .
Step 2: Examine the section of the graph between these intercepts. Since the graph dips below the x-axis between these values, $f(x) < 0$ in that interval.

Therefore, the function is negative between the roots, i.e., $1 < x < 7$ .

Thus, the solution to the problem is: $1 < x < 7$ .

Final Answer

$1 < x < 7$

Key Points to Remember

Essential concepts to master this topic

Graph Reading: Function is negative when curve lies below x-axis
Technique: Find x-intercepts first: x = 1 and x = 7
Check: Between intercepts graph dips below axis, so f(x) < 0 ✓

Common Mistakes

Avoid these frequent errors

Confusing where function is positive vs negative
Don't look at where the graph goes up or down = wrong regions! This confuses slope with sign of function values. Always check if the curve is above (positive) or below (negative) the x-axis.

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

How do I know when f(x) is negative just by looking at the graph?

Look for where the curve is below the x-axis. If you can draw a horizontal line from any point on the curve down to the x-axis, then f(x) < 0 at that x-value.

What do the x-intercepts tell me about the inequality?

X-intercepts are where f(x) = 0, which act as boundary points. The function changes sign (from positive to negative or vice versa) at these points.

Why is the answer 1 < x < 7 and not 1 ≤ x ≤ 7?

We need f(x) < 0 (strictly less than zero). At x = 1 and x = 7, the function equals zero, not less than zero, so we use < instead of ≤.

What if the graph touches but doesn't cross the x-axis?

If the graph just touches the x-axis at a point, that's still an x-intercept where f(x) = 0. The function doesn't change sign there, so check the intervals carefully.

How can I double-check my answer?

Pick a test point between your boundaries, like x = 4. Look at the graph: is f(4) above or below the x-axis? If below, then f(4) < 0, confirming your interval is correct!

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