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

Exercise #1

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 #2

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 #3

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 #4

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 #5

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 #1

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 #2

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 #3

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 #4

The slope of the function on the graph is 1.

What is the negative domain of the function?

(0, 1)(0, 1)(0, 1)xy

Video Solution

Step-by-Step Solution

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.

Answer

-1 > x

Exercise #5

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

Exercise #1

Calculate the slope of the line that passes through the points (4,1),(2,5) (4,1),(2,5) .

Video Solution

Step-by-Step Solution

Remember the formula for calculating a slope using points:

Now, replace the data in the formula with our own:

(51)(24)=42=2 \frac{(5-1)}{(2-4)}=\frac{4}{-2}=-2

Answer

-2

Exercise #2

What is the slope of a straight line that passes through the points (0,0),(8,2) (0,0),(-8,2) ?

Video Solution

Step-by-Step Solution

To solve the problem, remember the formula to find the slope using two points

 

Now, we replace the given points in the calculation:

 (02)(0(8)=28=14 \frac{(0-2)}{(0-(-8)}=\frac{-2}{8}=-\frac{1}{4}

Answer

14 -\frac{1}{4}

Exercise #3

Choose the correct answer for the function.

y=x+1 y=-x+1

Video Solution

Step-by-Step Solution

Let's start with option A

In a linear function, to check if the functions are parallel, you must verify if their slope is the same.

y = ax+b

The slope is a

In the original formula:

 y = -x+1

The slope is 1

In option A there is no a at all, which means it equals 1, which means the slope is not the same and the option is incorrect.

 

Option B:

To check if the function passes through the points, we will try to place them in the function:

-1 = -(-2)+1

-1 = 2+1

-1 = 3

The points do not match, and therefore the function does not pass through this point.

 

Option C:

We rearrange the function, in a way that is more convenient:

y = -1-x

y = -x-1

You can see that the slope in the function is the same as we found for the original function (-1), so this is the solution!

 

Option D:

When the slope is negative, the function is decreasing, as the slope is -1, the function is negative and this answer is incorrect.

Answer

The graph is parallel to the graph of function

y=1x y=-1-x

Exercise #4

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope

Exercise #5

For the function in front of you, the slope is?

XY

Video Solution

Answer

Negative slope