Select the correct answer.
Given the domain $\{-4,0,5\}$, what is the range for the relation $12 x+6 y=24 ?$
A. $\{12,4,-6\}$
B. $\{-4,4,14\}$
C. $\{-12,-4,6\}$
D. $\{2,4,9\}$
Answer (1)
Solve the equation for y : y = 4 − 2 x .
Substitute x = − 4 into the equation: y = 4 − 2 ( − 4 ) = 12 .
Substitute x = 0 into the equation: y = 4 − 2 ( 0 ) = 4 .
Substitute x = 5 into the equation: y = 4 − 2 ( 5 ) = − 6 . The range is { 12 , 4 , − 6 } .
Explanation
Understanding the Problem We are given the relation 12 x + 6 y = 24 and the domain { − 4 , 0 , 5 } . Our goal is to find the range of this relation. The range is the set of all possible y values that we get when we plug in the x values from the domain into the equation.
Solving for y First, let's solve the equation 12 x + 6 y = 24 for y in terms of x . We can do this by subtracting 12 x from both sides of the equation: 6 y = 24 − 12 x
Now, divide both sides by 6: y = 6 24 − 12 x = 4 − 2 x
Calculating y values Now that we have y in terms of x , we can substitute each value from the domain { − 4 , 0 , 5 } into the equation to find the corresponding y values.
For x = − 4 : y = 4 − 2 ( − 4 ) = 4 + 8 = 12
For x = 0 : y = 4 − 2 ( 0 ) = 4 − 0 = 4
For x = 5 : y = 4 − 2 ( 5 ) = 4 − 10 = − 6
Determining the Range The resulting set of y values is { 12 , 4 , − 6 } . Therefore, the range of the relation is { 12 , 4 , − 6 } .
Examples
Understanding relations and their ranges is crucial in many real-world scenarios. For example, consider a simple manufacturing process where the number of products made ( x ) affects the total cost ( y ). If you know the possible production levels (the domain), finding the range helps you determine the possible cost outcomes. This is also applicable in scenarios like currency conversion, where the amount in one currency ( x ) determines the equivalent amount in another currency ( y ), or in physics, where the time ( x ) affects the distance traveled ( y ) by an object.
