Part 1
In your own words, define the word “function.”
Give your own example of a function using a set of at least 4 ordered pairs. The domain will be any four integers between 0 and +10. The range will be any four integers between -12 and 5.Explain why your example models a function. This is extremely important for your learning.
Give your own example of at least four ordered pairs that does not model a function. The domain will be any four integers between 0 and +10. The range will be any four integers between -12 and +5.Explain why your example does not model a function.
Part 2
Select any two integers between -12 and +12 which will become solutions to a system of two equations.
Write two equations that have your two integers as solutions. Show how you built the equations using your integers. There are infinite possibilities.
Solve your system of equations by the addition/subtraction method. Make sure you show the necessary 5 steps.
Part 1:
In mathematics, a function is a relation between a set of inputs (the domain) and a set of outputs (the range) in which each input is associated with exactly one output. In simpler terms, a function takes an input and produces a single output.
Example of a function using four ordered pairs:
Let's consider the following set of ordered pairs: {(2, -12), (4, -5), (6, 0), (8, 5)}. In this example, the domain consists of the numbers 2, 4, 6, and 8, while the range consists of the numbers -12, -5, 0, and 5.
This set of ordered pairs represents a function because each input (domain value) has a unique corresponding output (range value). For every x-value in the domain, there is exactly one y-value in the range. No two ordered pairs share the same x-value with different y-values.
Example of a set of ordered pairs that does not model a function:
Let's consider the following set of ordered pairs: {(2, -12), (4, -5), (2, 0), (8, 5)}. In this example, the domain consists of the numbers 2, 4, 2, and 8, while the range consists of the numbers -12, -5, 0, and 5.
This set of ordered pairs does not represent a function because there is one repeated x-value (2) with different y-values (0 and -12). According to the definition of a function, each input should be associated with exactly one output. In this case, the input 2 has two distinct outputs, violating the definition.
Part 2:
Let the two integers be 5 and -7.
Equation 1: x + y = 5
To build this equation, I assigned the integer 5 to variable x and the integer -7 to variable y. I then added them together to get a sum of 5.
Equation 2: 2x - 3y = -7
To build this equation, I multiplied the integer 5 (assigned to x) by 2 and the integer -7 (assigned to y) by -3. I then subtracted the result of -21 from the left side of the equation to get -7.
Solving the system of equations using the addition/subtraction method:
Step 1: Multiply Equation 1 by 3 to eliminate the y terms:
3(x + y) = 3(5)
3x + 3y = 15
Step 2: Multiply Equation 2 by 2 to eliminate the x terms:
2(2x - 3y) = 2(-7)
4x - 6y = -14
Step 3: Add Equation 1 and Equation 2:
(3x + 3y) + (4x - 6y) = 15 + (-14)
7x - 3y = 1
Step 4: Multiply Equation 1 by 2 to eliminate the y terms:
2(x + y) = 2(5)
2x + 2y = 10
Step 5: Subtract Equation 2 from Equation 4:
(2x + 2y) - (7x - 3y) = 10 - 1
2x + 2y - 7x + 3y = 9
-5x + 5y = 9
The system of equations simplifies to:
-5x + 5y = 9
Part 1:
In mathematics, a function is a rule or a relationship that assigns each element from a set, called the domain, to a unique element in another set, called the range. It can also be thought of as a process that takes inputs from the domain and produces corresponding outputs in the range.
Example of a function:
Let's consider the following ordered pairs:
(2, -5), (4, -10), (6, 2), (8, -12)
In this example, the domain consists of the numbers 2, 4, 6, and 8, which are between 0 and 10. The range consists of the numbers -5, -10, 2, and -12, which are between -12 and 5. Each input value (in the domain) is uniquely associated with one output value (in the range). This means that for every x-value, there is only one y-value assigned to it. Therefore, this example models a function.
Example of non-function:
Let's consider the following ordered pairs:
(2, -5), (4, -10), (4, 2), (8, -12)
In this example, we can see that the x-value 4 from the domain is associated with two different y-values, i.e., -10 and 2. This violates the definition of a function, which requires each x-value to have only one corresponding y-value. Hence, this example does not model a function.
Part 2:
Let's select the integers 3 and -5 as solutions to the system of equations.
Equation 1:
To build the first equation, we can use the solution 3. Let's say we have an unknown variable 'x' and we want it to be equal to 3. Our equation can be written as:
x = 3
Equation 2:
To build the second equation, we can use the solution -5. Let's say we have another unknown variable 'y' and we want it to be equal to -5. Our equation can be written as:
y = -5
Now, let's solve this system of equations using the addition/subtraction method:
Step 1: Write the given equations:
x = 3
y = -5
Step 2: Rearrange the equations if necessary to make the variables line up:
x - 3 = 0
y + 5 = 0
Step 3: Multiply one or both equations by a number(s), so that when added/subtracted, one variable will cancel out:
Multiply equation 1 by -5:
-5(x - 3) = 0 → -5x + 15 = 0
Step 4: Add or Subtract the modified equations to eliminate one variable:
(-5x + 15) + (y + 5) = 0
-5x + y + 20 = 0
Step 5: Solve the equation for the remaining variable:
-5x + y + 20 = 0
-5x + y = -20
y = 5x - 20
So, the solution to the system of equations is:
x = 3 and y = 5x - 20