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question 2
which of the following exponential functions passes through the points (3, 40) and (9, 160)?

\\( f(x) = 40(2)^{\frac{x}{3}} \\)

\\( f(x) = 20(2)^{\frac{x}{6}} \\)

\\( f(x) = 40(2)^{\frac{x}{6}} \\)

\\( f(x) = 20(2)^{\frac{x}{4}} \\)

\\( f(x) = 20(2)^{\frac{x}{3}} \\)

\\( f(x) = 40(2)^{\frac{x}{2}} \\)

Explanation:

Step1: Test the first point (3, 40)

Let's substitute \( x = 3 \) into each function.

• For \( f(x)=40(2)^{x/3} \): \( f(3)=40(2)^{3/3}=40(2)^1 = 80

eq40 \)

• For \( f(x)=20(2)^{x/6} \): \( f(3)=20(2)^{3/6}=20(2)^{0.5}=20\sqrt{2}\approx28.28

eq40 \)

• For \( f(x)=40(2)^{x/6} \): \( f(3)=40(2)^{3/6}=40(2)^{0.5}=40\sqrt{2}\approx56.57

eq40 \)

• For \( f(x)=20(2)^{x/4} \): \( f(3)=20(2)^{3/4}=20\times2^{0.75}\approx20\times1.6818\approx33.64

eq40 \)

• For \( f(x)=20(2)^{x/3} \): \( f(3)=20(2)^{3/3}=20(2)^1 = 40 \). This matches the first point.

Step2: Test the second point (9, 160) with \( f(x)=20(2)^{x/3} \)

Substitute \( x = 9 \) into \( f(x)=20(2)^{x/3} \): \( f(9)=20(2)^{9/3}=20(2)^3 = 20\times8 = 160 \). This matches the second point.

Answer:

\( f(x) = 20(2)^{x/3} \) (the option: \( f(x)=20(2)^{x/3} \))