Question 104·200 Super-Hard SAT Math Questions·Advanced Math

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The exponential function $f$ is defined by

f(x)=\frac{48\cdot 12^{x}}{3^{2x-1}}.

The value of $f(2)$ is $k$ , where $k$ is a constant. Which choice is an equivalent form of $f$ that shows the value of $k$ as the coefficient or the base?

An exponential function changes by the same factor each time its input rises by $1$ . If a question asks for a form that shows a particular output as its coefficient, evaluate that output in Desmos and find the one-step multiplier. Then choose an exponent that is zero at the given input. A form can be equivalent without showing the requested value.

Hints

Hint 1
A coefficient is the number in front of a power. Evaluate $f(2)$ in Desmos first. If the power equals $1$ when $x=2$ , what must the coefficient be?
Hint 2
An exponential function uses the same multiplier for every one-unit increase in $x$ . Compare $f(3)$ with $f(2)$ , then look for an exponent that becomes $0$ when $x=2$ .

Step-by-step

Evaluate, find the multiplier, and shift the exponent
Step 1
Find the requested output
The problem calls $f(2)$ by the name $k$ . Type the given function into Desmos, then type $f(2)$ . Desmos prints $256$ , so the requested output is $k=256$ .
Step 2
Find the factor for each step in x
When $x$ rises by $1$ , the numerator's $12^x$ gains a factor of $12$ , while the denominator's $3^{2x-1}$ gains a factor of $3^2=9$ . So the one-step multiplier is $12/9$ . Add $f(3)/f(2)$ in Desmos; it prints $1.333\ldots$ . Tap the fraction button to see $\frac43$ .
Step 3
Make the coefficient equal f(2)
A nonzero base raised to the $0$ power equals $1$ . To show $f(2)$ as the coefficient, use the exponent $x-2$ : it is $0$ at $x=2$ , leaving $256\cdot1$ . Using $x+2$ would give an exponent of $4$ there, not $0$ . Make the exponent zero at the input whose output you want as the coefficient. With the $\frac43$ multiplier, the form is $f(x)=256\left(\frac43\right)^{x-2}$ . Choice A.

Question 104·200 Super-Hard SAT Math Questions·Advanced Math

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