question 1-7\nwrite an exponential function for the set of points.\n| x | f(x) |\n|----|----|\n| 0 | 27 |\n|…

question 1-7\nwrite an exponential function for the set of points.\n| x | f(x) |\n|----|----|\n| 0 | 27 |\n| 1 | 9 |\n| 2 | 3 |\n| 3 | 1 |\n| 4 | \\(\\frac{1}{3}\\) |\n\\(\\circ\\) \\(f(x) = \\left(\\frac{1}{3}\\right)^x\\)\n\\(\\circ\\) \\(f(x) = 3^x\\)\n\\(\\circ\\) \\(f(x) = 27\\left(\\frac{1}{3}\\right)^x\\)\n\\(\\circ\\) \\(f(x) = 27(3)^x\\)

question 1-7\nwrite an exponential function for the set of points.\n| x | f(x) |\n|----|----|\n| 0 | 27 |\n| 1 | 9 |\n| 2 | 3 |\n| 3 | 1 |\n| 4 | \\(\\frac{1}{3}\\) |\n\\(\\circ\\) \\(f(x) = \\left(\\frac{1}{3}\\right)^x\\)\n\\(\\circ\\) \\(f(x) = 3^x\\)\n\\(\\circ\\) \\(f(x) = 27\\left(\\frac{1}{3}\\right)^x\\)\n\\(\\circ\\) \\(f(x) = 27(3)^x\\)

Answer

Explanation:

Step1: Recall the form of exponential function

The general form of an exponential function is ( f(x) = ab^x ), where ( a ) is the initial value (when ( x = 0 )) and ( b ) is the base.

Step2: Find the value of ( a )

When ( x = 0 ), ( f(0) = 27 ). Substitute ( x = 0 ) into ( f(x) = ab^x ): ( f(0)=ab^0=a\times1=a ) So, ( a = 27 ).

Step3: Find the value of ( b )

We know ( a = 27 ), and let's use another point, say ( x = 1 ), ( f(1)=9 ). Substitute into ( f(x)=27b^x ): ( 9 = 27b^1 ) Solve for ( b ): ( b=\frac{9}{27}=\frac{1}{3} )

Step4: Write the function

Substitute ( a = 27 ) and ( b=\frac{1}{3} ) into ( f(x)=ab^x ): ( f(x)=27\left(\frac{1}{3}\right)^x )

Answer: ( f(x) = 27\left(\frac{1}{3}\right)^x ) (the option ( f(x)=27\left(\frac{1}{3}\right)^x ))