Function

What is a function?

A function is an equation that describes a specific relationship between XX and YY.
Every time we change XX, we get a different YY.

Linear function –

Looks like a straight line, XX is in the first degree.

Quadratic function –

Parabola, XX is in the square.

Practice Function

Examples with solutions for Function

Exercise #1

Look at the function shown in the figure.

When is the function positive?

xy-4-7

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 #2

Look at the linear function represented in the diagram.

When is the function positive?

–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333000

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

x > 2

Answer

x>2

Exercise #3

Solve the following inequality:

5x+8<9

Video Solution

Step-by-Step Solution

This is an inequality problem. The inequality is actually an exercise we solve in a completely normal way, except in the case that we multiply or divide by negative.

Let's start by moving the sections:

5X+8<9

5X<9-8

5X<1

We divide by 5:

X<1/5

And this is the solution!

 

Answer

x<\frac{1}{5}

Exercise #4

Solve the inequality:


5-3x>-10

Video Solution

Step-by-Step Solution

Inequality equations will be solved like a regular equation, except for one rule:

If we multiply the entire equation by a negative, we will reverse the inequality sign.

 

We start by moving the sections, so that one side has the variables and the other does not:

-3x>-10-5

-3x>-15

Divide by 3

-x>-5

Divide by negative 1 (to get rid of the negative) and remember to reverse the sign of the equation.

x<5

Answer

5 > x

Exercise #5

What is the solution to the following inequality?

10x43x8 10x-4≤-3x-8

Video Solution

Step-by-Step Solution

In the exercise, we have an inequality equation.

We treat the inequality as an equation with the sign -=,

And we only refer to it if we need to multiply or divide by 0.

 10x43x8 10x-4 ≤ -3x-8

We start by organizing the sections:

10x+3x48 10x+3x-4 ≤ -8

13x48 13x-4 ≤ -8

13x4 13x ≤ -4

Divide by 13 to isolate the X

x413 x≤-\frac{4}{13}

Let's look again at the options we were asked about:

Answer A is with different data and therefore was rejected.

Answer C shows a case where X is greater than413 -\frac{4}{13} , although we know it is small, so it is rejected.

Answer D shows a case (according to the white circle) where X is not equal to413 -\frac{4}{13} , and only smaller than it. We know it must be large and equal, so this answer is rejected.

 

Therefore, answer B is the correct one!

Answer

Exercise #6

Given the function of the figure.

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

xy2.253.5

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) (3.5,0) meaning when

x>3.5 below the X-axis

and when x < 3.5

above the X-axis.

In other words, the function is positive when x < 3.5

The function is negative when x>3.5

Answer

Positive x<3.5

Negative x>3.5

Exercise #7

Given the function of the graph.

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

xy7

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!

 

Answer

Positive 7 > x

Negative 7 < x

Exercise #8

Given the linear function of the drawing.

What is the negative domain of the function?

xy

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 #9

Given the linear function:

y=x4 y=x-4

What is the rate of change of the function?

Video Solution

Step-by-Step Solution

Let's remember that the rate of change equals the slope.

In this case, the slope is:

m=1 m=1

Answer

m=1 m=1

Exercise #10

Solve the inequality:

8x+a < 3x-4

Video Solution

Step-by-Step Solution

Solving an inequality equation is just like a normal equation. We start by trying to isolate the variable (X).

It is important to note that in this equation there are two variables (X and a), so we may not reach a final result.

 8x+a<3x-4

We move the sections

8x-3x<-4-a

We reduce the terms

5x<-4-a

We divide by 5

x< -a/5 -4/5

And this is the solution!

 

Answer

x < -\frac{1}{5}a-\frac{4}{5}

Exercise #11

Which best describes the function below?

y=23x y=2-3x

Video Solution

Step-by-Step Solution

Remember that the rate of change equals the slope.

In this function:

m=3 m=-3

Therefore, the function is decreasing.

Answer

The function is decreasing.

Exercise #12

Calculate the positive domain of the function shown in the figure:

(0, 4)(0, 4)(0, 4)xy

Video Solution

Step-by-Step Solution

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.

Answer

Not enough data

Exercise #13

Look at the function in the figure.

What is the positive domain of the function?

xy(0,a)(2a,0)

Video Solution

Step-by-Step Solution

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

Answer

2a < x

Exercise #14

What is the positive domain of the function shown in the graph below?

(0, 4)(0, 4)(0, 4)xy

Video Solution

Step-by-Step Solution

Since the entire function is above the X-axis, the function is always positive.

In other words, its positive domain will be all values of x x .

Answer

For all x x

Exercise #15

Given the function of the graph.

The slope is 1.5

What is the positive domain?

(0, -8)(0, -8)(0, -8)xy

Video Solution

Step-by-Step Solution

To find the domain of positivity, we need to find the point of intersection of the equation with the x-axis.

For this, we need to find the formula of the equation.

We know that a linear equation is constructed as follows:

Y=MX+B

m represents the slope of the line, which is given to us: 1.5

b represents the point of intersection of the line with the Y-axis, which can be extracted from the existing point on the graph, -8.

And therefore:

Y=1.5X-8

Now, we replace:

Y=0, since we are trying to find the point of intersection with the X-axis.

0=1.5X-8
8=1.5X
5.3333 = X

We reveal that the point of intersection with the X-axis is five and one third (5.333)

Now, as we know that the slope is positive and the function is increasing, we can conclude that the domain of positivity is when the x values are less than five and one third.

That is:

5.333>X

And this is the solution!

Answer

5\frac{1}{3}>x