Positive and Negativity of a Linear Function

Positivity and Negativity of a Linear Function

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?

1 - Positivity and Negativity of a Linear Function

Detailed explanation

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Test yourself on positivity and negativity!

Look at the function shown in the figure.

When is the function positive?

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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 : $X$

A - Positivity and Negativity of a Linear Function

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, when $Y<0$

We will take the equation equal to $Y$

$4X-2$

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

Exercise #1

Look at the linear function represented in the diagram.

When is the function positive?

Video Solution

Step-by-Step Solution

The function is positive when it is above the X-axis.

Let's note that the intersection point of the graph with the X-axis is:

$(2,0)$ meaning any number greater than 2:

$x > 2$

Answer

$x>2$

Exercise #2

Look at the function shown in the figure.

When is the function positive?

Video Solution

Step-by-Step Solution

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

Answer

$-4 > x$

Exercise #3

Look at the function graphed below.

What are the areas of positivity and negativity of the function?

Video Solution

Step-by-Step Solution

Let's remember that the function is positive when it is above the X-axis. The function is negative when it is below the X-axis.

Let's note that the intersection point of the graph with the X-axis is:

$(3.5,0)$

This means that when $x>3.5$ , it is below the X-axis and when $x < 3.5$ , it is above the X-axis.

In other words, the function is positive when $x < 3.5$ and the function is negative when $x>3.5$ .

Answer

^{Positive $x<3.5$}

^{Negative $x>3.5$}

Exercise #4

Given the linear function of the drawing.

What is the negative domain of the function?

Video Solution

Step-by-Step Solution

The function is negative when it is below the Y-axis.

Note that the graph always remains above the X-axis, meaning it is always positive.

Answer

The always positive function

Exercise #5

Given the function of the graph.

What are the areas of positivity and negativity of the function?

Video Solution

Step-by-Step Solution

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!