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For the functions in this problem, we have that:

a) The function is exponential, and the table is completed as follows.

  • x = 0: y = 1.
  • x = 1: y = 5.
  • x = 2: y = 25.
  • x = 3: y = 125.
  • x = 4: y = 625.
  • x = 5: y = 3125.

b) The function is linear, and the table is completed as follows.

  • x = 0: y = 0.
  • x = 1: y = 5.
  • x = 2: y = 10.
  • x = 3: y = 15.
  • x = 4: y = 20.
  • x = 5: y = 25.

c) The function is exponential and given by:

[tex]y = -16\left(\frac{1}{2}\right)^x[/tex]

What is an exponential function?

An exponential function is modeled by:

[tex]y = ab^x[/tex]

In which:

  • a is the initial value.
  • b is the rate of change.

A function with initial value 1 and where the y-value is computed by multiplied the previous y-value by 5, as in item a, is an exponential function with parameters a = 1 and b = 5, hence the rule is:

[tex]y = 5^x[/tex]

Then:

  • x = 0: y = 1.
  • x = 1: y = 5.
  • x = 2: y = 25.
  • x = 3: y = 125.
  • x = 4: y = 625.
  • x = 5: y = 3125.

What is a proportional relationship?

A proportional relationship is a function in which the output variable is given by the input variable multiplied by a constant of proportionality, that is:

y = kx

In which k is the constant of proportionality.

A proportional relationship is also classified as linear with a intercept of 0. For item b, since each y-value is the x-value multiplied by 5, the rule is:

y = 5x.

Hence the table is:

  • x = 0: y = 0.
  • x = 1: y = 5.
  • x = 2: y = 10.
  • x = 3: y = 15.
  • x = 4: y = 20.
  • x = 5: y = 25.

For item c, when x changes by 1, y is always multiplied by 1/2, and due to the multiplication, the function is exponential and given by:

[tex]y = -16\left(\frac{1}{2}\right)^x[/tex]

The -16 is because when x = 0, y = -16.

More can be learned about functions at https://brainly.com/question/25537936

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