Input 3 5 4 7 output 6 11 11 21 write an equation, in function notation, that would fit the inputs and outputs in the table. show work that proves your equation is true for the given data.

If we graph the given points then they appear to be in the shape of cubic function so we can use standard formula of cubic function which is

Plug the given points to get four equations

like first point (3,6) so plug x=3 and y=6 into above equation, we get:

or

6=27a+9b+3c+d

Same way by plugging other points form table, we get total four equations as follows:

6=27a+9b+3c+d, 11=125a+25b+5c+d, 11=64a+16b+4c+d, 21=343a+49b+7c+d

Now we just need to solve these equations for a, b, c and d.

Solving them to get values of a, b, c and d, we get:

Now plug them into

So the required equation will be:

Hence required equation in function notation can be written as

Now to prove that above function is correct, we just graph the given points from table and the obtained function.

After graphing the given points from table and the obtained function, We see that points lie on the graph of

Which proves that our equation is correct.

option 2 is correct.

step-by-step explanation:

we are given two inequality equation

x+y> 5 and -3x+2y< -2

first we draw the graph of both function using t-table method

equation: x+y> 5

for x=0, y=5

for x=-5, y=0

table:

x   :   0     5

y   :   5     0

test point (0,0) , 0> 5

false

equation: -3x+2y< -2

for x=0, y=-1

for x=2, y=4

table:

x   :   0     2

y   : -1       4

test point (0,0) , 0< -2

true

now we draw the graph. see the attachment for graph.

common region would be solution set.

option 2 is correct.