consider the exponential function ( f(x)=3left(\frac{1}{3}\right)^{x} ) and its graph. which statements are…

consider the exponential function ( f(x)=3left(\frac{1}{3}\right)^{x} ) and its graph. which statements are true for this function and graph? select three options. the initial value of the function is ( \frac{1}{3} ). the base of the function is ( \frac{1}{3} ). the function shows exponential decay. the function is a stretch of the function ( f(x)=left(\frac{1}{3}\right)^{x} ). the function is a shrink of the function ( f(x)=3^{x} ).

consider the exponential function ( f(x)=3left(\frac{1}{3}\right)^{x} ) and its graph. which statements are true for this function and graph? select three options. the initial value of the function is ( \frac{1}{3} ). the base of the function is ( \frac{1}{3} ). the function shows exponential decay. the function is a stretch of the function ( f(x)=left(\frac{1}{3}\right)^{x} ). the function is a shrink of the function ( f(x)=3^{x} ).

Answer

Brief Explanations:

  • For the function (f(x)=a\cdot b^{x}), (a) is the initial value. Here (a = 3), so the statement "The initial value of the function is (\frac{1}{3})" is false.
  • For the function (f(x)=3(\frac{1}{3})^{x}), the base (b=\frac{1}{3}), so the statement "The base of the function is (\frac{1}{3})" is true.
  • Since (0 < b=\frac{1}{3}<1), the function (y = 3(\frac{1}{3})^{x}) shows exponential decay, so the statement "The function shows exponential decay" is true.
  • The function (y = 3(\frac{1}{3})^{x}) can be compared to (y = (\frac{1}{3})^{x}). When we multiply (y = (\frac{1}{3})^{x}) by (a = 3>1), it is a vertical stretch of (y = (\frac{1}{3})^{x}), so the statement "The function is a stretch of the function (f(x)=(\frac{1}{3})^{x})" is true.
  • The function (y = 3(\frac{1}{3})^{x}=3\cdot3^{-x}=3^{1 - x}), and (y = 3^{x}) has a base (b = 3>1) (exponential growth), while (y=3^{1 - x}) is different from a simple shrink of (y = 3^{x}), so the statement "The function is a shrink of the function (f(x)=3^{x})" is false.

Answer:

The base of the function is (\frac{1}{3}), The function shows exponential decay, The function is a stretch of the function (f(x)=(\frac{1}{3})^{x})