Question

# Determine whether each graph represents a function ?

Hint:

### We can check if the graph is a function by the vertical line test. A graph represents a function if any vertical line in the xy plane cuts the graph at maximum one point. We use this method to check if each of the given graphs represent a function.

## The correct answer is: not a function

*Step by step solution:*

We consider the first graph.

We can observe that any vertical line drawn on the graph cuts the line at exactly one point

Hence, this graph represents a function.

The second graph is

Again, we can observe that any vertical line drawn cuts the graph at exactly one point.

Hence, this graph is also a function.

Finally, consider the third graph.

If we draw a vertical line at the origin, that is, the y-axis, we can see that it cuts the graph at two points.

Thus, this graph is not a function.

An equation of the form y = f(x) is called a function if there is a unique value of y for every value of x. In other words, every value of x must have one value of y. When we draw a vertical, it cuts x axis at one point, we take that point to be x. If the vertical line cuts the graph at two points, then it gives two values of y for one value of x. This does not represent a function.

### Related Questions to study

### Choose the negative adjectives starting with ' u '

Explanation-Negative adjectives are the word that explains / pronounce negatively.

### Choose the negative adjectives starting with ' u '

Explanation-Negative adjectives are the word that explains / pronounce negatively.

### Describe the possible values of x.

- Step-by-step explanation:

- Given:

a = x + 11, b = 2x + 10, and c = 5x - 9.

- Step 1:
- First check validity.

According to triangle inequality theorem,

c - b < a < b + c,

(5x – 9) – (2x + 10) < x + 11 < (2x + 10) + (5x – 9)

3x - 19 < x + 11 < 7x + 1

First consider,

x + 11 < 7x + 1,

11 – 1 < 7x - x

10 < 6x

< x,

1.6 < x

Now, consider,

3x - 19 < x + 11

3x - x < 11 + 19

2x < 30

x < ,

x < 15

therefore,

1.6 < x < 15

- Final Answer:

### Describe the possible values of x.

- Step-by-step explanation:

- Given:

a = x + 11, b = 2x + 10, and c = 5x - 9.

- Step 1:
- First check validity.

According to triangle inequality theorem,

c - b < a < b + c,

(5x – 9) – (2x + 10) < x + 11 < (2x + 10) + (5x – 9)

3x - 19 < x + 11 < 7x + 1

First consider,

x + 11 < 7x + 1,

11 – 1 < 7x - x

10 < 6x

< x,

1.6 < x

Now, consider,

3x - 19 < x + 11

3x - x < 11 + 19

2x < 30

x < ,

x < 15

therefore,

1.6 < x < 15

- Final Answer:

### Write the solutions to the given equation.

Rewrite them as the linear-quadratic system of equations and graph them to solve.

A quadratic equation is when the polynomial has a degree two. A graph is a geometrical representation of an equation.

We are asked to solve the equation graphically by arranging them in a linear-quadratic equation.

Step 1 of 2:

The given equation is

Re arranging them, we have:

Step 2 of 2:

Graph the quadratic equation:

The solution is .

Note:

A quadratic equation can be solved using different identities and even simplifying them.

### Write the solutions to the given equation.

Rewrite them as the linear-quadratic system of equations and graph them to solve.

A quadratic equation is when the polynomial has a degree two. A graph is a geometrical representation of an equation.

We are asked to solve the equation graphically by arranging them in a linear-quadratic equation.

Step 1 of 2:

The given equation is

Re arranging them, we have:

Step 2 of 2:

Graph the quadratic equation:

The solution is .

Note:

A quadratic equation can be solved using different identities and even simplifying them.

### Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

5 inches, 12 inches

- Hints:
- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c

c < a + b

- while finding possible lengths of third side use below formula

- Step-by-step explanation:

- Given:

a = 5 inches, b = 12 inches.

- Step 1:
- Find length of third side.

c < a + b

∴ c < 5 + 12

c < 17

- Step 2:

b – a < c < a + b

12 – 5 < c < 5 + 12

7 < c < 17

Hence, all numbers between 7 and 17 will be the length of third side.

- Final Answer:

### Describe the possible lengths of the third side of the triangle given the lengths of the other two sides.

5 inches, 12 inches

- Hints:
- Triangle inequality theorem
- According to this theorem, in any triangle, sum of two sides is greater than third side,
- a < b + c

c < a + b

- while finding possible lengths of third side use below formula

- Step-by-step explanation:

- Given:

a = 5 inches, b = 12 inches.

- Step 1:
- Find length of third side.

c < a + b

∴ c < 5 + 12

c < 17

- Step 2:

b – a < c < a + b

12 – 5 < c < 5 + 12

7 < c < 17

Hence, all numbers between 7 and 17 will be the length of third side.

- Final Answer:

### Identify the common noun in the given sentence

"The mice are afraid of a cat"

Explanation-Mice Words which are used as names of persons, animals, places or things are called Common nouns

### Identify the common noun in the given sentence

"The mice are afraid of a cat"

Explanation-Mice Words which are used as names of persons, animals, places or things are called Common nouns

### How many solutions does the given system of equations have?

y=x & y=x^{2}

If the LHS or RHS of two equations are equal, then we could equate them. An equation is a mathematical statement, where we equate two equal expressions.

We are asked to find how many solutions the given equations have.

Step 1 of 2:

The equations are:

Here, the LHS of the equations are equal. So, we have:

.

Step 2 of 2:

Solving them, we get:

Corresponding to these values, we get:

Thus, we have two solutions for the given set of equations.

Note:

We could find the number of solutions for a given set of equations, by analyzing the power of the equation.

### How many solutions does the given system of equations have?

y=x & y=x^{2}

If the LHS or RHS of two equations are equal, then we could equate them. An equation is a mathematical statement, where we equate two equal expressions.

We are asked to find how many solutions the given equations have.

Step 1 of 2:

The equations are:

Here, the LHS of the equations are equal. So, we have:

.

Step 2 of 2:

Solving them, we get:

Corresponding to these values, we get:

Thus, we have two solutions for the given set of equations.

Note:

We could find the number of solutions for a given set of equations, by analyzing the power of the equation.

### How can you determine whether the relationship between side length and area is a function?

If it is a function, say whether it is linear or non linear function ?

*Step by step solution:*

Let us denote the length by x.

Let us denote the area by y.

From the table, we can observe that there is a different and unique value of y for every x.

This gives that the given relationship is a function.

Next, we draw the graph of the function to check its linearity.

We construct the following table

x |
0 |
1 |
2 |
3 |
4 |
5 |

y |
0 |
1 |
4 |
9 |
16 |
25 |

We plot the above points in the following graph.

Clearly, the graph is not a straight line. Hence the given relationship between length and area is not a linear function.

### How can you determine whether the relationship between side length and area is a function?

If it is a function, say whether it is linear or non linear function ?

*Step by step solution:*

Let us denote the length by x.

Let us denote the area by y.

From the table, we can observe that there is a different and unique value of y for every x.

This gives that the given relationship is a function.

Next, we draw the graph of the function to check its linearity.

We construct the following table

x |
0 |
1 |
2 |
3 |
4 |
5 |

y |
0 |
1 |
4 |
9 |
16 |
25 |

We plot the above points in the following graph.

Clearly, the graph is not a straight line. Hence the given relationship between length and area is not a linear function.

### A 10,000 gallon swimming pool needs to be emptied. Exactly 2000 gallons have already been pumped out of the pool and into the tanker. How can you determine how long it will take to pump all of the water into the tanker. 720 gallons per hour can be emptied from the pool.

*Hint:*

We are given the rate at which water can be emptied from the pool. We are given the initial value of how much water has already been pumped. This information helps us to find a relationship between the quantity of water in the pool and the time taken to pump the water out of the pool. We construct a linear function and we find the time taken to pump out the water from the equation

*Step by step solution*:

Let the time (in hours) taken be denoted by x and let the quantity of water in the pool be denoted by y.

First, we find a linear function representing the relationship between x and y

Given,

The rate at which the water is pumped= 720 gallons per hour

Quantity of water already pumped out = 2000 gallons

The total quantity of water in the pool = 10000 gallons

From the slope intercept form of an equation y = mx + c, where m is slope and c is the y-intercept,

And using the values m=720 and c =2000, we get

We need to pump out 10000 gallons of water, so putting y= 10000 in the above equation, we get

10000 = 720x + 2000

Next, we solve for x,

Dividing by 720 throughout, we have

Thus, we will need 11.11 hours to empty the pool.

*Note:*

We can also solve this problem without the use of functions. There is 10000 gallons of water and 2000 gallons of water is already emptied. So there is (10000-2000=) 8000 gallons of water. It is given that the rate at which water is pumped is 720 gallons per hour. So the time taken is hours

### A 10,000 gallon swimming pool needs to be emptied. Exactly 2000 gallons have already been pumped out of the pool and into the tanker. How can you determine how long it will take to pump all of the water into the tanker. 720 gallons per hour can be emptied from the pool.

*Hint:*

We are given the rate at which water can be emptied from the pool. We are given the initial value of how much water has already been pumped. This information helps us to find a relationship between the quantity of water in the pool and the time taken to pump the water out of the pool. We construct a linear function and we find the time taken to pump out the water from the equation

*Step by step solution*:

Let the time (in hours) taken be denoted by x and let the quantity of water in the pool be denoted by y.

First, we find a linear function representing the relationship between x and y

Given,

The rate at which the water is pumped= 720 gallons per hour

Quantity of water already pumped out = 2000 gallons

The total quantity of water in the pool = 10000 gallons

From the slope intercept form of an equation y = mx + c, where m is slope and c is the y-intercept,

And using the values m=720 and c =2000, we get

We need to pump out 10000 gallons of water, so putting y= 10000 in the above equation, we get

10000 = 720x + 2000

Next, we solve for x,

Dividing by 720 throughout, we have

Thus, we will need 11.11 hours to empty the pool.

*Note:*

We can also solve this problem without the use of functions. There is 10000 gallons of water and 2000 gallons of water is already emptied. So there is (10000-2000=) 8000 gallons of water. It is given that the rate at which water is pumped is 720 gallons per hour. So the time taken is hours

### Identify proper noun in the given sentence

Anubha is a wonderful lady?

Explanation - Proper noun refers to specific person, place, animal or thing

### Identify proper noun in the given sentence

Anubha is a wonderful lady?

Explanation - Proper noun refers to specific person, place, animal or thing

### A line can intersect a parabola maximum of ___.

A parabola is a plane curve which is mirror symmetrical and is approximately U shaped. A line is a straight one dimensional figure having no thickness and extended infinitely to both directions.

We are asked to find how many times maximum could a line intersect a parabola.

Step 1 of 1:

We know that a line is extended to infinity in both the direction and a parabola is a U shaped symmetric structure. So, the maximum number of times a line could intersect a parabola is two times.

Note:

A parabola could be opened downwards, upwards or sideways. The center may not always be at the origin.

### A line can intersect a parabola maximum of ___.

A parabola is a plane curve which is mirror symmetrical and is approximately U shaped. A line is a straight one dimensional figure having no thickness and extended infinitely to both directions.

We are asked to find how many times maximum could a line intersect a parabola.

Step 1 of 1:

We know that a line is extended to infinity in both the direction and a parabola is a U shaped symmetric structure. So, the maximum number of times a line could intersect a parabola is two times.

Note:

A parabola could be opened downwards, upwards or sideways. The center may not always be at the origin.

### Choose the simple subject in the given sentence.

"My little brother broke his fingers".

Explanation - Simple subject is the single word that is the subject of the verb

### Choose the simple subject in the given sentence.

"My little brother broke his fingers".

Explanation - Simple subject is the single word that is the subject of the verb

### Choose the correct spelling word, to form a logic sentence

"The is in the center of our solar system”

Explanation - Homophones are words that sound the same, but have different spelling and meaning.

### Choose the correct spelling word, to form a logic sentence

"The is in the center of our solar system”

Explanation - Homophones are words that sound the same, but have different spelling and meaning.

### Identify the type of Sentence: " She smiles when she is happy".

Explanation - An independent idea and a dependent idea in one sentence

### Identify the type of Sentence: " She smiles when she is happy".

Explanation - An independent idea and a dependent idea in one sentence

### Sketch the following line on a graph: y = 8/9 x - 6

*Hint:*

To plot the graph of an equation, first we make a table of points satisfying that equation. Then we draw an x-axis and y-axis on the graph. After that we scale both the axis according the values we get in the table. Lastly, we plot the points from the table on the graph and join them to get the required curve.

*Step by step solution:*

The given equation is

First we make a table of points satisfying the equation.

Putting x = 0 in the above equation, we will get, y = -6

Similarly, putting x = 9, in the above equation, we get y = 2

Continuing this way, we have

For x = -9, we get y = -14

For x = 18, we get y = 10

For x = -18, we get y = -22

Making a table of all these points, we have

Now we plot these points on the graph.

After plotting the points, we join them with a line to get the graph of the equation.

*Note:*

We can find the tabular values for any points of x and then plot them on the graph. But we usually choose values for which calculating y is easier. This makes plotting the graph simpler. We can also find the values by putting different values of y in the equation to get different values for x. Either way, we need points satisfying the equation to plot its graph

### Sketch the following line on a graph: y = 8/9 x - 6

*Hint:*

To plot the graph of an equation, first we make a table of points satisfying that equation. Then we draw an x-axis and y-axis on the graph. After that we scale both the axis according the values we get in the table. Lastly, we plot the points from the table on the graph and join them to get the required curve.

*Step by step solution:*

The given equation is

First we make a table of points satisfying the equation.

Putting x = 0 in the above equation, we will get, y = -6

Similarly, putting x = 9, in the above equation, we get y = 2

Continuing this way, we have

For x = -9, we get y = -14

For x = 18, we get y = 10

For x = -18, we get y = -22

Making a table of all these points, we have

Now we plot these points on the graph.

After plotting the points, we join them with a line to get the graph of the equation.

*Note:*

We can find the tabular values for any points of x and then plot them on the graph. But we usually choose values for which calculating y is easier. This makes plotting the graph simpler. We can also find the values by putting different values of y in the equation to get different values for x. Either way, we need points satisfying the equation to plot its graph

### Sketch the following line on a graph: y = 3/2 x + 4

*Hint:*

To plot the graph of an equation, first we make a table of points satisfying that equation. Then we draw an x-axis and y-axis on the graph. After that we scale both the axis according the values we get in the table. Lastly, we plot the points from the table on the graph and join them to get the required curve.

*Step by step solution:*

The given equation is

First we make a table of points satisfying the equation.

Putting x = 0 in the above equation, we will get, y = 4

Similarly, putting x = 2, in the above equation, we get y = 7

Continuing this way, we have

For x = -2, we get y = 1

For x = 4, we get y = 10

For x = -4, we get y = -2

Making a table of all these points, we have

Now we plot these points on the graph.

After plotting the points, we join them with a line to get the graph of the equation.

*Note:*

We can find the tabular values for any points of x and then plot them on the graph. But we usually choose values for which calculating y is easier. This makes plotting the graph simpler. We can also find the values by putting different values of y in the equation to get different values for x. Either way, we need points satisfying the equation to plot its graph

### Sketch the following line on a graph: y = 3/2 x + 4

*Hint:*

To plot the graph of an equation, first we make a table of points satisfying that equation. Then we draw an x-axis and y-axis on the graph. After that we scale both the axis according the values we get in the table. Lastly, we plot the points from the table on the graph and join them to get the required curve.

*Step by step solution:*

The given equation is

First we make a table of points satisfying the equation.

Putting x = 0 in the above equation, we will get, y = 4

Similarly, putting x = 2, in the above equation, we get y = 7

Continuing this way, we have

For x = -2, we get y = 1

For x = 4, we get y = 10

For x = -4, we get y = -2

Making a table of all these points, we have

Now we plot these points on the graph.

After plotting the points, we join them with a line to get the graph of the equation.

*Note:*

We can find the tabular values for any points of x and then plot them on the graph. But we usually choose values for which calculating y is easier. This makes plotting the graph simpler. We can also find the values by putting different values of y in the equation to get different values for x. Either way, we need points satisfying the equation to plot its graph