Positive and Negative Domains: Graphical representation with data

Examples with solutions for Positive and Negative Domains: Graphical representation with data

Exercise #1

Find all values of x x

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

XXXYYY-6-6-6-10-10-10-2-2-2

Step-by-Step Solution

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

The graph of the function f(x) f(x) shows it is below the x-axis in the interval from x=10 x = -10 to x=2 x = -2 . Between these points, f(x) 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 x = -10 and x=2 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 f(x) < 0 is 10<x<2-10 < x < -2.

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

Answer

-10 < x < -2

Exercise #2

Find all values of x x

where f\left(x\right) > 0 .

XXXYYY-11-11-11-1-1-1-6-6-6

Step-by-Step Solution

To solve the given problem using the graph, we need to determine the intervals along the x-axis where the quadratic function f(x) f(x) is positive, based on its x-intercepts x=11 x = -11 and x=1 x = -1 as shown on the graph.

  • Step 1: Identify the x-intercepts from the graph: x=11 x = -11 and x=1 x = -1 .
  • Step 2: Interpret the graph of the quadratic function. Since it is a parabola opening upwards and touches the x-axis at x=11 x = -11 and x=1 x = -1 , these are points where the quadratic changes sign.
  • Step 3: Determine the intervals: The graph is above the x-axis (positive) between the x-intercepts because the parabola is opening upwards. Therefore, the function is positive for 11<x<1 -11 < x < -1 .

The conclusion is that the quadratic function f(x) f(x) is greater than zero in the interval 11<x<1 -11 < x < -1 .

Therefore, the correct answer is 11<x<1\mathbf{-11 < x < -1}.

Answer

-11 < x < -1

Exercise #3

Find all values of x x

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

XXXYYY-11-11-11-1-1-1-6-6-6

Step-by-Step Solution

To determine where the function f(x) f(x) is less than 0, observe the graphical representation:

  • The roots are located at x=11 x = -11 , x=6 x = -6 , and x=1 x = -1 . These are the x-values where the function intersects the x-axis.
  • Considering the general behavior of quadratic functions, the function is negative between the outer roots unless it passes through below x-axis at multiple roots due to shape.

The given graph suggests the function dips below the x-axis between x=11 x = -11 and x=1 x = -1 , passing through x=6 x = -6 .

After analyzing the intervals:

  • The interval to the left: x<11 x < -11
  • The interval to the right: x>1 x > -1

Therefore, values of x x for which the function f(x) f(x) is less than 0 are x>1 x > -1 or x<11 x < -11 .

The correct choice is: x>1 x > -1 or x<11 x < -11

Answer

x > -1 or x < -11

Exercise #4

Find all values of x x

where f\left(x\right) > 0 .

XXXYYY-4-4-4

Step-by-Step Solution

In this problem, we are tasked with determining the values of x x for which the function f(x) f(x) is positive. We have been provided a graphical representation of the function, and we will use this graph to find our solution.

1. Restate the problem: We need to find all values of x x where the function f(x) f(x) is greater than zero, based on its graphical representation. 2. Identify key information: The graph is typically that of some function f(x) f(x) . The graph shows points and lines that illustrate where the function is above and below the x-axis. Points or curves on or above the x-axis indicate positive values. 3. Potential approach: Analyze where the graph is above the x-axis. 5. The most appropriate approach is to visually inspect the graph to identify when the curve is above the x-axis. 6. Steps needed: - Identify any turning points or intersections with the x-axis. - Determine the segments of the x-axis where the function is above it. 8. Simplify the inspection by focusing on intervals separated by intersections with the x-axis. 9. Consider that the function might only touch the x-axis at specific points, like at roots, and analyze behavior around these points.

Based on the graph, we observe the following behavior of the function f(x) f(x) :

  • The function intersects the x-axis at x=4 x = -4 . This indicates a potential root or turning point where the function transitions from positive to negative or vice versa.
  • From the graph, it appears that the function is above the x-axis on both sides of x=4 x = -4 , except exactly at x=4 x = -4 , where it touches the x-axis.

Hence, the function f(x) f(x) is positive for x>4 x > -4 and for x<4 x < -4 . Note that exactly at x=4 x = -4 , the function is zero, not positive.

Therefore, the solution is: x>4 x > -4 or x<4 x < -4 .

In conclusion, the function f(x) f(x) is positive for these values of x x , except the point where it touches the x-axis.

The corresponding choice given the problem's options is:

x > -4 or x < -4

Answer

x > -4 or x < -4

Exercise #5

Find all values of x

where f(x) < 0 .

XXXYYY-4-4-4

Step-by-Step Solution

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

The process to follow is:

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

From this analysis, the function f(x) f(x) is negative for all x x except at x=4 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 x > -4 or x<4 x < -4 .

Answer

x > -4 or x < -4

Exercise #6

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

XXXYYY000-2-2-2

Step-by-Step Solution

Based on the graph provided, we can see the entire function lies below the x-axis. Thus, there is no interval where f(x)>0 f(x) > 0 .

To solve this problem, here's what we observed:

  • Visual inspection of the graph reveals that it never crosses the x-axis from below.
  • Consequently, the function remains non-positive for all x-values visible, indicating it's non-positive overall within the range observable.

Therefore, the function has no domain where it is positive. Therefore, the solution is:

The function has no domain where it is positive

Answer

The function has no domain where it is positive

Exercise #7

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 Solution

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

  • Step 1: Identify the x-intercepts of the graph, which are x=1 x = 1 and x=7 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 f(x) < 0 in that interval.

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

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

Answer

1 < x < 7

Exercise #8

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

XXXYYY222888555

Step-by-Step Solution

To solve for when f(x) < 0 on the graph, we follow these steps:

  • Step 1: Locate the x-intercepts, where the curve intersects the x-axis. These intercepts are x=2 x = 2 and x=8 x = 8 .

  • Step 2: Analyze the sections determined by these intercepts. The graph is below the x-axis to the left of x=2 x = 2 and to the right of x=8 x = 8 .

By visually inspecting the graph, it is evident that:

  • The function f(x) f(x) is below the x-axis (i.e., negative) for x < 2 and x > 8 .

Therefore, the solution to the problem is that the graph of the function is negative for x > 8 or x < 2 .

Answer

x > 8 or x < 2

Exercise #9

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

XXXYYY-2-2-2222000

Step-by-Step Solution

The problem is asking us to identify for which x x values f(x)>0 f(x) > 0 based on the graph provided, which seems to depict a quadratic function. Let's go step-by-step:

First, we need to determine the points where the function intersects the x-axis, which are the roots of the function. The graph shows these intersections at x=2 x = -2 and x=2 x = 2 . These are the points where the function is equal to zero, f(x)=0 f(x) = 0 .

Next, we observe the overall shape of the graph to understand where f(x)>0 f(x) > 0 (i.e., where the graph is above the x-axis). Typically for a quadratic function, which is a parabola, the parabola will be above the x-axis outside the roots if it opens upwards, and between the roots if it opens downwards, given that a(xx1)(xx2)=0 a(x - x_1)(x - x_2) = 0 with root analysis on a>0 a > 0 .

In the provided graph, the parabola appears to open upwards. Therefore, the function f(x) f(x) is positive when x x is less than the smaller root, 2 -2 , or greater than the larger root, 2 2 . This is a typical behavior for a quadratic function which opens upwards, where it takes negative values inside the range of its roots and positive values outside.

Conclusively, f(x)>0 f(x) > 0 for the intervals where x<2 x < -2 or x>2 x > 2 .

Therefore, the solution to the problem is x>2 x > 2 or x<2 x < -2 .

Answer

x > 2 or x < -2

Exercise #10

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

XXXYYY-2-2-2222000

Step-by-Step Solution

To solve the problem of finding for which x x values the function f(x)<0 f(x) < 0 , we proceed as follows:

First, we observe the provided graph of the function. Our goal is to identify the intervals on the x x -axis where the curve of the function is below the line y=0 y = 0 (the x-axis). These intervals represent where the function f(x) f(x) takes negative values.

Upon examining the graph, we notice that:

  • The curve descends below the x-axis starting once it crosses x=2 x = -2 .
  • It continues below the x-axis until it reaches x=2 x = 2 .
  • Therefore, the function is negative between these two points.

Based on the graph, the interval where f(x)<0 f(x) < 0 is from x=2 x = -2 to x=2 x = 2 . Thus, the correct mathematical statement for the values of x x where f(x)<0 f(x) < 0 is 2<x<2 -2 < x < 2 .

The correct choice from the options given is \(\text{2<x<2 -2 < x < 2 }\).

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

Answer

-2 < x < 2

Exercise #11

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

XXXYYY-3-3-3333000

Step-by-Step Solution

First, we examine the provided graph of the quadratic function f(x) f(x) . The graph clearly shows the x-intercepts (where the function crosses the x-axis) at x=3 x = -3 and x=3 x = 3 .

Since the quadratic function appears to be a standard parabola opening upwards, the portion of the graph between these two x-intercepts will be above the x-axis, which means that f(x)>0 f(x) > 0 in this interval.

The intervals to the left of x=3 x = -3 and to the right of x=3 x = 3 will be where the graph lies below the x-axis, meaning f(x)<0 f(x) < 0 in those regions.

Thus, the graph shows that the function f(x) f(x) is positive between x=3 x = -3 and x=3 x = 3 . Therefore, the solution to the problem is:

3<x<3 -3 < x < 3

Answer

-3 < x < 3

Exercise #12

Look at the function graphed below.

Find all values of x x

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

000-3-3-3333XY

Video Solution

Answer

x > 3 or x < -3

Exercise #13

Find all values of x

where f\left(x\right) > 0 .

XXXYYY-6-6-6-10-10-10-2-2-2

Video Solution

Answer

x > -2 or x > -10

Exercise #14

Find all values of x x

where f\left(x\right) > 0 .

XXXYYY555

Video Solution

Answer

x < 5 or x > 5

Exercise #15

Find all values of x x

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

XXXYYY888

Video Solution

Answer

x < 8 or 8 < x

Exercise #16

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

XXXYYY222888555

Video Solution

Answer

2 < x < 8