The function is positive when it is above the X axis when Y<0
The function is negative when it is below the X axis as Y>0
When we are asked what the domains of positivity of the function are, we are actually being asked in which values of X the function is positive: when it is above the X axis.
In which values of X does the function obtain positive Y values?
When we are asked what the domain of negativity of the function is, we are actually being asked in which values of X the function is negative: when it is below the X axis.
In which values of X does the function obtain negative Y values?
Now, we will see an example in which we find the domain of positivity and negativity of the function by solving an inequality:
Let's see an example in which we find the domains of positivity and negativity of the function through a graph:
Let's observe the point of intersection of the function with the axis :X
Given the function: y=4x−2
What is the domain of positivity and what is the domain of negativity of the function?
Let's remember that when we are asked about the domain of positivity, we are asked in which values of X, the values of Y are positive.
Therefore, when Y>0
We will take the equation equal to Y
4X−2
And check when it is greater than 0
4X−2>0
We solve the inequality:
4X>2
X>0.5
The domain of positivity of the line is:
X>0.5
Now let's remember that when we are asked about the domain of negativity, we are asked in which values of X, the values of Y are negative.
Therefore, whenY<0
We will take the equation equal toY
4X−2
And check when it is less than 0
4X−2<0
We solve the inequality:
4X<2
X<0.5
The domain of negativity of the line is:
X<0.5
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Examples and exercises with solutions on the positivity and negativity of a linear function
examples.example_title
Look at the function shown in the figure.
When is the function positive?
examples.explanation_title
The function we see is a decreasing function,
Because as X increases, the value of Y decreases, creating the slope of the function.
We know that this function intersects the X-axis at the point x=-4
Therefore, we can understand that up to -4, the values of Y are greater than 0, and after -4, the values of Y are less than zero.
Therefore, the function will be positive only when
X < -4
examples.solution_title
-4 > x
examples.example_title
Given the function of the graph.
What are the areas of positivity and negativity of the function?
examples.explanation_title
When we are asked what the domains of positivity of the function are, we are actually being asked at what values of X the function is positive: it is above the X-axis.
At what values of X does the function obtain positive Y values?
In the given graph, we observe that the function is above the X-axis before the point X=7, and below the line after this point. That is, the function is positive when X>7 and negative when X<7,
And this is the solution!
examples.solution_title
Positive 7 > x
Negative 7 < x
examples.example_title
Calculate the positive domain of the function shown in the figure:
examples.explanation_title
The domains of positivity and negativity are determined by the point of intersection of the function with the X-axis, so the Y values are greater or less than 0.
We are given the information of the intersection with the Y-axis, but not of the point of intersection with the X-axis,
Furthermore, there is no data about the function itself or the slope, so we do not have the ability to determine the point of intersection with the X-axis,
And so in the domains of positivity and negativity.
examples.solution_title
Not enough data
examples.example_title
Look at the function in the figure.
What is the positive domain of the function?
examples.explanation_title
Positive domain is another name for the point from which the x values are positive and not negative.
From the figure, it can be seen that the function ascends and passes through the intersection point with the X-axis (where X is equal to 0) at point 2a.
Therefore, it is possible to understand that from the moment X is greater than 2a, the function is in the domains of positivity.
Therefore, the function is positive when:
2a < x
examples.solution_title
2a < x
examples.example_title
The slope of the function on the graph is 1.
What is the negative domain of the function?
examples.explanation_title
To answer the question, let's first remember what the "domain of negativity" is,
The domain of negativity: when the values of Y are less than 0.
Note that the point given to us is not the intersection point with the X-axis but with the Y-axis,
That is, at this point the function is already positive.
The point we are looking for is the second one, where the intersection with the X-axis occurs.
The function we are looking at is an increasing function, as can be seen in the diagram and the slope (a positive slope means that the function is increasing),
This means that if we want to find the point, we have to find an X that is less than 0
Now let's look at the solutions:
Option B and Option D are immediately ruled out, since in them X is greater than 0.
We are left with option A and C.
Option C describes a situation in which, as X is less than 0, the function is negative,
Remember that we know the slope is 1,
Which means that for every increase in X, Y also increases in the same proportion.
That is, if we know that when (0,1) the function is already positive, and we want to lower Y to 0,
X also decreased in the same value. If both decrease by 1, the resulting point is (0,-1)
From this we learn that option C is incorrect and option A is correct.
Whenever X is less than -1, the function is negative.
examples.solution_title
-1 > x
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