Question

Topic: Recognizing Linear and Exponential and Quadratic Functions.

Answer 1

Answer:

  Recursive: f(1) = 1; f(n) = f(n-1)/3

  Equation: f(x) = 5/3·(1/3)^(x-1)

Step-by-step explanation:

You know this is an exponential function because sequential lines in the table have a common ratio of f(x) values:

  f(2)/f(1) = (5/9)/(5/3) = 3/9 = 1/3

  f(3)/f(2) = (5/27)/(5/9) = 9/27 = 1/3

and it continues like that.

Besides telling you the function is exponential, it also tells you that each term is 1/3 of the previous term.

Recursive formula

The value of f(1) is read from the table. For x = 1, that value is ...

  f(1) = 5/3

As we noticed above, each term is 1/3 the previous term, so the recursive relation is ...

  f(n) = (1/3)·f(n-1)

Equation

The general form of the equation for a geometric (exponential) relation is ...

  f(x) = f(1)·r^(x-1) . . . . . . where f(1) is the first term, and r is the common ratio

For this function, we have f(1) = 5/3 and r = 1/3, so the equation is ...

  f(x) = 5/3·(1/3)^(x-1)

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Additional comment

When given values of x that count from 1, we often express the equation in terms of (x-1). That equation can be simplified so the exponent is x.

  f(x) = 5·(1/3)^x