Is g = {(1, 1), (2, 3), (3, 5), (4, 7)} a function? Justify. If this is described by the
relation, g (x) = αx + β, then what values should be assigned to α and β?
One of the most important properties of a straight line is in how it angles away from the horizontal.
This concept is reflected in something called the "slope" of the line
For straight lines, slope is constant (always the same).
You can graph the points and "count" the vertical changes and horizontal changes to use in the formula:
á = slope = Vertical change / horizontal change
á = slope = [ g (n) - g ( n-1 ) ] / [ x (n) - x ( n-1 )
In this case :
á = [ g (2) - g ( 1 ) ] / [ x (2) - x ( 1 ) = ( 3 - 1 ) / ( 2 - 1 ) = 2 / 1 = 2
á = [ g (3) - g ( 2 ) ] / [ x (3) - x ( 2 ) = ( 5 - 3 ) / ( 3 - 2 ) = 2 / 1 = 2
á = [ g (4) - g (3 ) ] / [ x (4) - x ( 3 ) = ( 7 - 5 ) / ( 4 - 3 ) = 2 / 1 = 2
á = slope = 2
When you know á you can put any point in your equation. For example x = 3 , g = 5
g (x) = áx + â
5 = 2 * 3 + â
5 = 6 + â Subtract 6 to both sides
5 - 6 = 6 + â - 6
- 1 = â
â = - 1
g (x) = áx + â
g (x) = 2x - 1
á = Greek letter alpha
â = Greek letter beta
To determine whether g = {(1, 1), (2, 3), (3, 5), (4, 7)} is a function, we need to check if each input (x) has a unique output (y).
In this case, for each x-value, there is only one corresponding y-value. For example, when x = 1, y = 1; when x = 2, y = 3; when x = 3, y = 5; and when x = 4, y = 7.
Therefore, g = {(1, 1), (2, 3), (3, 5), (4, 7)} is indeed a function.
Now, if g(x) is described by the relation g(x) = αx + β, where α and β are constants, we need to find the values for α and β.
To find α and β, we can substitute the values from the given function into the equation g(x) = αx + β and solve for the constants.
Let's take the first pair of (x, y): (1, 1)
Substituting these values into the equation, we have:
1 = α * 1 + β
Similarly, we can substitute the other pairs of (x, y) into the equation, resulting in three additional equations:
3 = α * 2 + β
5 = α * 3 + β
7 = α * 4 + β
Now, we have a system of equations in terms of the unknowns α and β.
To solve this system, we can use various methods, such as substitution or elimination. However, I will use the method of substitution for simplicity.
Starting with the first equation, 1 = α + β, we can isolate β:
β = 1 - α
Next, we substitute this expression for β into the other equations and solve for α.
3 = α * 2 + 1 - α
5 = α * 3 + 1 - α
7 = α * 4 + 1 - α
Simplifying these equations:
3 = α + 1
5 = α + 1
7 = α + 1
From these equations, we find that α = 2.
Now, substituting this value of α back into the expression for β:
β = 1 - 2 = -1
Therefore, the values assigned to α and β are α = 2 and β = -1, respectively.
So, the relation g(x) = αx + β can be written as g(x) = 2x - 1.