The graph in the image appears to be a cubic function with five marked points. Here’s a summary of the graph and the statements provided:

Question

Graph Summary:
The function decreases from left to right until it reaches a local minimum.
It then increases to a local maximum before decreasing again.
Statements:
f(x) > 0 over the interval (1, ∞): True. The function is above the x-axis for x-values greater than 1.
f(x) < 0 over the interval (-∞, -1): True. The function is below the x-axis for x-values less than -1.
f(x) > 0 over the interval (-∞, -1): False. This contradicts statement 2.
f(x) > 0 over the interval (-∞, ∞): False. There are intervals where f(x) < 0.
The graph visually represents where the function’s output values are positive or negative relative to intervals on the x-axis. This can be useful in understanding real-world phenomena modeled by such functions or in solving inequalities.

Answer 1

Based on the summary and the statements provided about the cubic function, let's analyze each statement and confirm the validity of the conclusions drawn:

1. Statement: f(x) > 0 over the interval (1, ∞)

   • Analysis: This statement is noted as true, meaning the function is above the x-axis for x-values greater than 1.

2. Statement: f(x) < 0 over the interval (-∞, -1)

   • Analysis: This statement is also marked as true, indicating the function is below the x-axis for x-values less than -1.

3. Statement: f(x) > 0 over the interval (-∞, -1)

   • Analysis: This statement is false, as it directly contradicts statement 2. The function is not above the x-axis in that interval.

4. Statement: f(x) > 0 over the interval (-∞, ∞)

   • Analysis: This statement is false. Since the function dips below the x-axis in certain intervals (specifically for the ranges mentioned), it cannot be true that f(x) > 0 for all x-values.

In summary, statements 1 and 2 are supported by the graph, while statements 3 and 4 are clearly incorrect based on the function's behavior described in the graph. The analysis of where the function is positive or negative provides insight into the function's overall shape and can help in applications such as solving inequalities or understanding real-world scenarios modeled by the cubic function.

Understanding the intervals where the function takes positive or negative values can also be crucial for identifying roots, potential maximum and minimum points, and behavior of the function in different contexts, such as in physics or economics where such functions might arise.

Answer 2

Use the graphing tool to determine the true statements regarding the represented function. Check all that apply.

f(x) > 0 over the interval (1, infinity).
f(x) 0 over the interval [1, infinity).
f(x) 0 over the interval (-infinity, 1].
f(x) > 0 over the interval (-infinity, 1).
f(x) > 0 over the interval (-infinity, infinity).

which statements are true?

Answer 3

To determine the truth of the statements regarding the function based on the graph you described earlier, let's analyze each statement one by one, considering the characteristics of a cubic function as you've outlined:

1. Statement: f(x) > 0 over the interval (1, ∞)

   • If the function is above the x-axis for x-values greater than 1, this statement is True.

2. Statement: f(x) ≥ 0 over the interval [1, ∞)

   • This statement includes 1 in its interval and means that the function could be zero at x = 1, but if it's only positive for x-values greater than 1, the truth of this statement would depend on whether the function is also non-negative (zero or positive) at x = 1. If the graph shows the function touches or is above the x-axis at x = 1, the statement is True; otherwise, it is False.

3. Statement: f(x) ≤ 0 over the interval (-∞, 1]

   • If the function is below the x-axis for x-values less than 1 (considering x=1 for equality), this statement is True. If it touches the x-axis (is zero) at x = 1, then for the interval (-∞, 1], to be true, it must be confirmed that the function does not exceed 0 for values less than or equal to 1.

4. Statement: f(x) > 0 over the interval (-∞, 1)

   • If the function is below the x-axis for values less than 1, this statement would be False.

5. Statement: f(x) > 0 over the interval (-∞, ∞)

   • Since you indicated that the function takes negative values in certain intervals, this statement is False.

Summary of Truth Values:

• True Statements:

  • f(x) > 0 over the interval (1, ∞)
  • f(x) ≥ 0 over the interval [1, ∞) (if f(1) = 0 or f(1) > 0)
  • f(x) ≤ 0 over the interval (-∞, 1] (if it confirms the function behavior in that interval)

• False Statements:

  • f(x) > 0 over the interval (-∞, 1)
  • f(x) > 0 over the interval (-∞, ∞)

Ultimately, you would need to clarify the behavior of the cubic function at the point x = 1 to fully validate statement 2. If you have access to the graph, you can directly check the function values at 1 to confirm the exact truths.

Answer 4

f(x) > 0 over the interval (1, infinity). and f(x) 0 over the interval (-infinity, 1].

A 2-column table with 6 rows. The first column is labeled x with entries negative 3, negative 2, negative 1, 0, 1, 2. The second column is labeled f of x with entries 50, 0, negative 6, negative 4, negative 6, 0.
Use the table to complete the statements.

The x-intercepts shown in the table are
and
.

The y-intercept shown in the table is
.

Answer 5

Given the information from the table you provided, we can analyze the entries:

• x values: -3, -2, -1, 0, 1, 2
• f(x) values: 50, 0, -6, -4, -6, 0

Determining the Intercepts

1. X-Intercepts: The x-intercepts of a function occur where \( f(x) = 0 \). According to the table:

   • \( f(-2) = 0 \) (at x = -2)
   • \( f(2) = 0 \) (at x = 2)

   Therefore, the x-intercepts shown in the table are:

   • -2 and 2

2. Y-Intercept: The y-intercept of a function occurs where \( x = 0 \). According to the table:

   • \( f(0) = -4 \)

   Therefore, the y-intercept shown in the table is:

   • -4

Summary of the Statements

• The x-intercepts shown in the table are -2 and 2.
• The y-intercept shown in the table is -4.