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Please check the explanation.

Step-by-step explanation:

Here are some of the rules using which we can easily figure out the true nature of a function.

Linear Function:

The constant change in x and y

Quadratic Function:

The constant change in x

The constant 2nd differences in y

Exponential Function:

The constant change in x

The constant ratio in y

Determining the nature of function

Given the table

x    -3     -2     -1     0      1       2

y      32    16     8     4      2      1

It is clear that there is a constant change in x

i.e. -2-(-3) = 1, -1-(-2) = 1, 0 - (-1) = 1, 1-0 = 1, 2-1 = 1

Also there is a constant ratio in y

16/32 = 1/2, 8/16 = 1/2, 4/8 = 1/2, 2/4=1/2, 1/2 = 1/2

Therefore, the given function is the exponential function.

Determining the equation

We know the exponential function is of the form

y = abˣ

Taking the point (0, 4) and substitute in the function

4 = ab⁰

as b⁰ = 1, so

4 = a(1)

a = 4

substituting the point (1, 2) and a = 4 in the function

y = abˣ

2 = 4b¹

as b¹  = b, so

2 = 4b

b = 2/4

b = 1/2

Now, substituting a = 4, and b = 1/2 in the function

y = abˣ

y = 4(1/2)ˣ

Therefore, the equation of the function is:

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step-by-step explanation: