Question

test information description instructions multiple attempts this test allows multiple attempts. force completion this test can be saved and resumed later. your answers are saved automatically. question completion status: moving to another question will save this response. question 9 which of the following exponential functions passes through the points (8,80) and (12,160)? ( f(x) = 20(2)^{x/4} ) ( f(x) = 40(2)^{x/2} ) ( f(x) = 20(2)^{x/6} ) ( f(x) = 20(2)^{x/3} ) ( f(x) = 40(2)^{x/3} ) ( f(x) = 40(2)^{x/6} )

Explanation:

Step1: Test the first point (8,80) in each function

• For \( f(x) = 20(2)^{x/4} \): Substitute \( x = 8 \), we get \( 20(2)^{8/4}=20(2)^2 = 20\times4 = 80 \). This works for (8,80). Let's check the second point.
• For \( f(x) = 40(2)^{x/2} \): Substitute \( x = 8 \), \( 40(2)^{8/2}=40(2)^4 = 40\times16 = 640

eq80 \). Eliminate.

• For \( f(x) = 20(2)^{x/6} \): Substitute \( x = 8 \), \( 20(2)^{8/6}=20(2)^{4/3}\approx20\times2.5198\approx50.396

eq80 \). Eliminate.

• For \( f(x) = 20(2)^{x/3} \): Substitute \( x = 8 \), \( 20(2)^{8/3}=20(2)^{2 + 2/3}=20\times4\times2^{2/3}\approx20\times4\times1.5874\approx126.99

eq80 \). Eliminate.

• For \( f(x) = 40(2)^{x/3} \): Substitute \( x = 8 \), \( 40(2)^{8/3}=40\times2^{2 + 2/3}=40\times4\times2^{2/3}\approx40\times4\times1.5874\approx253.99

eq80 \). Eliminate.

• For \( f(x) = 40(2)^{x/6} \): Substitute \( x = 8 \), \( 40(2)^{8/6}=40(2)^{4/3}\approx40\times2.5198\approx100.79

eq80 \). Eliminate.

Step2: Test the second point (12,160) in the remaining function \( f(x) = 20(2)^{x/4} \)

Substitute \( x = 12 \) into \( f(x) = 20(2)^{x/4} \), we get \( 20(2)^{12/4}=20(2)^3 = 20\times8 = 160 \). This works for (12,160).

Answer:

\( f(x) = 20(2)^{x/4} \) (the first option)