determine the range of each relation for the domain {-3,-2,-1,0,1,2,3}
a) y=-3x+4
b)y=2x^2+5
c)y=1/x-4
d) y=-5(x+1)^2-4
To determine the range of each relation for the given domain {-3, -2, -1, 0, 1, 2, 3}, we substitute each value of the domain into the respective equation and calculate the corresponding y-values.
a) y = -3x + 4:
Let's substitute the values of x from the domain into the equation:
For x = -3: y = -3*(-3) + 4 = 13
For x = -2: y = -3*(-2) + 4 = 10
For x = -1: y = -3*(-1) + 4 = 7
For x = 0: y = -3*0 + 4 = 4
For x = 1: y = -3*1 + 4 = 1
For x = 2: y = -3*2 + 4 = -2
For x = 3: y = -3*3 + 4 = -5
The range of this relation is {-5, -2, 1, 4, 7, 10, 13}.
b) y = 2x^2 + 5:
Now, let's substitute the values of x from the domain into this equation:
For x = -3: y = 2*(-3)^2 + 5 = 23
For x = -2: y = 2*(-2)^2 + 5 = 9
For x = -1: y = 2*(-1)^2 + 5 = 7
For x = 0: y = 2*0^2 + 5 = 5
For x = 1: y = 2*1^2 + 5 = 7
For x = 2: y = 2*2^2 + 5 = 13
For x = 3: y = 2*3^2 + 5 = 23
The range of this relation is {5, 7, 9, 13, 23}.
c) y = 1/x - 4:
Let's substitute the values of x from the domain into this equation:
For x = -3: y = 1/(-3) - 4 = -4 1/3
For x = -2: y = 1/(-2) - 4 = -4.5
For x = -1: y = 1/(-1) - 4 = -5
For x = 0: y = 1/0 - 4 = undefined (division by zero is undefined)
For x = 1: y = 1/1 - 4 = -3
For x = 2: y = 1/2 - 4 = -3.5
For x = 3: y = 1/3 - 4 = -3 2/3
The range of this relation is {-5, -4 1/3, -4.5, -3 2/3, -3.5}.
d) y = -5(x+1)^2 - 4:
Substitute the values of x from the domain into this equation:
For x = -3: y = -5*(-3+1)^2 - 4 = -24
For x = -2: y = -5*(-2+1)^2 - 4 = -19
For x = -1: y = -5*(-1+1)^2 - 4 = -4
For x = 0: y = -5*(0+1)^2 - 4 = -9
For x = 1: y = -5*(1+1)^2 - 4 = -9
For x = 2: y = -5*(2+1)^2 - 4 = -24
For x = 3: y = -5*(3+1)^2 - 4 = -49
The range of this relation is {-49, -24, -19, -9, -4}.
To determine the range of each relation for the given domain {-3,-2,-1,0,1,2,3}, we need to calculate the corresponding outputs (values of y) for each value of x in the domain.
a) y = -3x + 4
For each x in the given domain, we substitute the value of x and solve for y:
When x = -3: y = -3(-3) + 4 = 13
When x = -2: y = -3(-2) + 4 = 10
When x = -1: y = -3(-1) + 4 = 7
When x = 0: y = -3(0) + 4 = 4
When x = 1: y = -3(1) + 4 = 1
When x = 2: y = -3(2) + 4 = -2
When x = 3: y = -3(3) + 4 = -5
So, the range of relation (a) is {-5, -2, 1, 4, 7, 10, 13}.
b) y = 2x^2 + 5
For each x in the given domain, we substitute the value of x and solve for y:
When x = -3: y = 2(-3)^2 + 5 = 23
When x = -2: y = 2(-2)^2 + 5 = 13
When x = -1: y = 2(-1)^2 + 5 = 7
When x = 0: y = 2(0)^2 + 5 = 5
When x = 1: y = 2(1)^2 + 5 = 7
When x = 2: y = 2(2)^2 + 5 = 13
When x = 3: y = 2(3)^2 + 5 = 23
So, the range of relation (b) is {5, 7, 13, 23}.
c) y = 1/(x - 4)
For each x in the given domain, we substitute the value of x and solve for y:
When x = -3: y = 1/(-3 - 4) = -1/7
When x = -2: y = 1/(-2 - 4) = -1/6
When x = -1: y = 1/(-1 - 4) = -1/5
When x = 0: y = 1/(0 - 4) = -1/4
When x = 1: y = 1/(1 - 4) = -1/3
When x = 2: y = 1/(2 - 4) = -1/2
When x = 3: y = 1/(3 - 4) = -1
So, the range of relation (c) is {-1/7, -1/6, -1/5, -1/4, -1/3, -1/2, -1}.
d) y = -5(x + 1)^2 - 4
For each x in the given domain, we substitute the value of x and solve for y:
When x = -3: y = -5(-3 + 1)^2 - 4 = -9
When x = -2: y = -5(-2 + 1)^2 - 4 = -5
When x = -1: y = -5(-1 + 1)^2 - 4 = -4
When x = 0: y = -5(0 + 1)^2 - 4 = -9
When x = 1: y = -5(1 + 1)^2 - 4 = -24
When x = 2: y = -5(2 + 1)^2 - 4 = -49
When x = 3: y = -5(3 + 1)^2 - 4 = -84
So, the range of relation (d) is {-84, -49, -24, -9, -5, -4, -9}.
These are the ranges of each relation for the given domain.
To find the range, substitute the values from the domain.
Example: y = -3x + 4
y = -3(-3) + 4 = 9 + 4 = 13
y = -3(-2) + 4 = 6 + 4 = 10
So, we are finding the range of 13 and 10 s0 far