Complete the following table so that f(x) represents a linear function and g(x)
represents an exponential function,

x     f(x)    g(x)

-2    6       1/9

-1     10      1/3

0     14      1

1      18      3

2     22     9

Step-by-step explanation:

Model for a linear function:

f(x) = ax + b

Using the points (-2, 6) and (0,14), we can find the values of a and b:

6 = -2a + b

14 = 0a + b

b = 14

From the first equation:

6 = -2a + 14

2a = 8

a = 4

So we have that f(x) = 4x + 14

Now we find the values of f(x) for x = -1, x = 1 and x = 2:

f(-1) = -4 + 14 = 10

f(1) = 4 + 14 = 18

f(2) = 8 + 14 = 22

Model for exponencial equation:

g(x) = a*b^x

Using the points (-2, 1/9) and (-1,1/3), we can find the values of a and b:

1/9 = a*b^(-2)

1/3 = a*b^(-1)

Dividing the second equation by the first, we have:

b = (1/3) / (1/9) = 3

From the second equation:

1/3 = a * b^(-1)

1/3 = a * 1/3

a = 1

So we have that:

g(x) = 3^x

Now we find the values of g(x) for x = 0, x = 1 and x = 2:

g(0) = 3^0 = 1

g(1) = 3^1 = 3

g(2) = 3^2 = 9

the direct variation says that:

then the equation is of the form is given by:

            where k is the constant variation.

from the given table:

consider any value of x and m(x)=y

x = 2 and m(x)=y = 3

substitute these value in [1]; we have;

divide both sides by 2 we get;

⇒

check:

put x = -1 and m(x) = -1.5

substitute in the equation:

-1.5 = -1.5                   true

therefore, the function m(x) represents a direct variation

graph of this function as shown below: