Which statements are true of functions? Check all that apply. try to choose the correct

Question

All functions have a dependent variable.
All functions have an independent variable.
The range of a function includes its domain.
A vertical line is an example of a functional relationship.
A horizontal line is an example of a functional relationship.
Each output value of a function can correspond to only one input value.

Answer 1

Let's analyze each statement regarding functions:

1. All functions have a dependent variable.

   • True. In a function, the dependent variable's value depends on the value of the independent variable.

2. All functions have an independent variable.

   • True. Functions typically have an independent variable, which is the variable that you can choose freely and for which the function defines a corresponding output.

3. The range of a function includes its domain.

   • False. The domain refers to all possible input values (independent variable), while the range refers to all possible output values (dependent variable). The range does not include the domain, unless in very special cases.

4. A vertical line is an example of a functional relationship.

   • False. A vertical line fails the vertical line test, meaning it can intersect the graph at multiple points, which means it does not represent a function.

5. A horizontal line is an example of a functional relationship.

   • True. A horizontal line passes the vertical line test as it has a constant output value for any input, representing a valid function.

6. Each output value of a function can correspond to only one input value.

   • False. In a function, each input value must correspond to exactly one output value, but multiple input values can correspond to the same output value.

So, the true statements are:

• All functions have a dependent variable.
• All functions have an independent variable.
• A horizontal line is an example of a functional relationship.

Answer 2

Which represents a function?

A 2-column table with 5 rows. The first column is labeled x with entries negative 5, negative 3, negative 3, 0, 5. The second column is labeled y with entries 10, 5, 4, 0, negative 10.

StartSet (negative 8, negative 2), (negative 4, 1), (0, negative 2), (2, 3), (4, negative 4) EndSet

A 2-column table with 5 rows. The first column is labeled x with entries negative 2, negative 1, 0, 0, 1. The second column is labeled y with entries negative 3, negative 2, negative 1, 0, negative 1.

StartSet (negative 12, 4), (negative 6, 10), (negative 4, 15), (negative 8, 18), (negative 12, 24) EndSet

Answer 3

To determine which of the given representations constitutes a function, we need to check if each input (x-value) corresponds to exactly one unique output (y-value).

1. First representation (2-column table):

   • x values: -5, -3, -3, 0, 5
   • y values: 10, 5, 4, 0, -10
   • The x-value -3 corresponds to two different y-values (5 and 4).
   • Conclusion: Not a function.

2. Second representation (set of ordered pairs):

   • (-8, -2), (-4, 1), (0, -2), (2, 3), (4, -4)
   • Each x-value is unique: -8, -4, 0, 2, and 4.
   • Conclusion: This is a function.

3. Third representation (2-column table):

   • x values: -2, -1, 0, 0, 1
   • y values: -3, -2, -1, 0, -1
   • The x-value 0 corresponds to two different y-values (-1 and 0).
   • Conclusion: Not a function.

4. Fourth representation (set of ordered pairs):

   • (-12, 4), (-6, 10), (-4, 15), (-8, 18), (-12, 24)
   • The x-value -12 corresponds to two different y-values (4 and 24).
   • Conclusion: Not a function.

Summary:

• The only representation that is a function is the second representation (the set of ordered pairs: { (-8, -2), (-4, 1), (0, -2), (2, 3), (4, -4) }).

Answer 4

Consider a function that describes how a particular car’s gas mileage depends on its speed. What would be an appropriate domain for this function?

0 to 100 miles per hour
0 to 50 miles per gallon
times from 0 to 10 minutes
times from –10 to 10 minutes

Answer 5

In the context of a function that describes how a particular car's gas mileage depends on its speed, we are interested in the range of speeds the car can realistically operate at.

Given the options you provided:

1. 0 to 100 miles per hour - This represents a reasonable range of speeds for a car. Most cars operate safely and efficiently at speeds between 0 and 100 mph.

2. 0 to 50 miles per gallon - This is a measure of fuel efficiency (miles per gallon), not speed. This would represent a range for the output of the function, not for the input (speed).

3. times from 0 to 10 minutes - This time range does not pertain to the speed of the car. Again, this would correlate to a different aspect of driving but does not inform us about speed.

4. times from –10 to 10 minutes - This also refers to time rather than speed and includes negative time, which is not meaningful in this context.

Appropriate Domain:

The most appropriate domain for this function is 0 to 100 miles per hour. This range captures realistic and relevant speeds for a car while analyzing its gas mileage.

Answer 6

What are the possible values of x and y for two distinct points, (5, –2) and (x, y), to represent a function?



The value of x can be
.



The value of y can be
.

Answer 7

To determine the possible values of \( x \) and \( y \) for the point \( (x, y) \) such that the pair \( (5, -2) \) and \( (x, y) \) represent a function when graphed, we need to ensure that the \( x \)-values (inputs) are unique.

Given Point:

• The first point is \( (5, -2) \).

Considerations:

1. Distinct Points: If \( (5, -2) \) and \( (x, y) \) are to be distinct points, then \( x \) cannot be equal to \( 5 \).

2. Function Representation: Each \( x \) value in a function must correspond to exactly one \( y \) value. This means if \( x \) takes on any value other than \( 5 \), then \( y \) can be any value.

Possible Values:

• The value of \( x \) can be: Any real number except \( 5 \) (e.g., \( x \in \mathbb{R}, x \neq 5 \)).

• The value of \( y \) can be: Any real number (e.g., \( y \in \mathbb{R} \)).

So, to summarize:

• The value of \( x \) can be any real number except \( 5 \).
• The value of \( y \) can be any real number.

Answer 8

Emma is an artist and has created seven paintings. She plans to paint three additional paintings each week. What statements are true of this scenario? Check all that apply.

The independent variable is represented by the output values.
The dependent variable is the total number of paintings.
The total number of paintings depends on the number of weeks that have passed.
This scenario is a function because for each week that passes, there will be exactly one total number of paintings.
The number of weeks is the output value.

Answer 9

Let's analyze each statement regarding Emma's painting scenario.

1. The independent variable is represented by the output values.

   • False. In this context, the independent variable is typically the number of weeks that have passed, which is an input rather than an output value.

2. The dependent variable is the total number of paintings.

   • True. The total number of paintings is dependent on the number of weeks that have passed, as she is adding more paintings each week.

3. The total number of paintings depends on the number of weeks that have passed.

   • True. The total number of paintings increases as the weeks progress, so it indeed depends on the number of weeks.

4. This scenario is a function because for each week that passes, there will be exactly one total number of paintings.

   • True. For each unit increase in weeks, there is one consistent increase in the total number of paintings (3 additional paintings per week), making this a function.

5. The number of weeks is the output value.

   • False. The number of weeks is the input (independent variable), and the total number of paintings is the output (dependent variable).

Summary of True Statements:

• The dependent variable is the total number of paintings.
• The total number of paintings depends on the number of weeks that have passed.
• This scenario is a function because for each week that passes, there will be exactly one total number of paintings.