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question 1
which of the following exponential functions passes through the points (3,80) and (9,320) ?
$f(x)=40(2)^{x/6}$
$f(x)=40(2)^{x/3}$
$f(x)=20(2)^{x/6}$
$f(x)=20(2)^{x/4}$
$f(x)=20(2)^{x/3}$
$f(x)=40(2)^{x/2}$

Explanation:

Step1: Recall exponential function form

The general form of an exponential function is \( f(x) = ab^{kx} \), but here the options are in the form \( f(x)=C(2)^{x/n} \), so we can test each option by plugging in \( x = 3 \) and \( x = 9 \) to see if we get \( f(3)=80 \) and \( f(9)=320 \).

Step2: Test option \( f(x)=40(2)^{x/6} \)

For \( x = 3 \): \( f(3)=40(2)^{3/6}=40(2)^{1/2}=40\sqrt{2}\approx56.57
eq80 \). So this is not correct.

Step3: Test option \( f(x)=40(2)^{x/3} \)

For \( x = 3 \): \( f(3)=40(2)^{3/3}=40\times2 = 80 \).
For \( x = 9 \): \( f(9)=40(2)^{9/3}=40\times2^{3}=40\times8 = 320 \).
This satisfies both points. Let's check other options quickly to be sure.

Step4: Test option \( f(x)=20(2)^{x/6} \)

For \( x = 3 \): \( f(3)=20(2)^{3/6}=20\sqrt{2}\approx28.28
eq80 \). Incorrect.

Step5: Test option \( f(x)=20(2)^{x/4} \)

For \( x = 3 \): \( f(3)=20(2)^{3/4}\approx20\times1.6818\approx33.64
eq80 \). Incorrect.

Step6: Test option \( f(x)=20(2)^{x/3} \)

For \( x = 3 \): \( f(3)=20(2)^{3/3}=20\times2 = 40
eq80 \). Incorrect.

Step7: Test option \( f(x)=40(2)^{x/2} \)

For \( x = 3 \): \( f(3)=40(2)^{3/2}=40\times2.828\approx113.14
eq80 \). Incorrect.

Answer:

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