Question 194·200 Super-Hard SAT Math Questions·Advanced Math

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For the exponential function $f$ defined by

f(x)=\frac{4\cdot 2^{5x-6}\cdot 9^{x+2}}{6^{3x-4}},

the value of $f(3)$ is $k$ , where $k$ is a constant. Which choice gives an equivalent form of $f$ that shows the value of $k$ as the coefficient or the base?

An exponential function can be written as $A b^{x-h}$ . The coefficient $A$ multiplies the power, and the base $b$ is the multiplier for each one-unit increase in $x$ . At $x=h$ , the exponent is $0$ , so the coefficient equals the output at that input. Evaluate that output in Desmos, then compare outputs one unit apart to find the base. Matching the coefficient alone doesn't establish equivalence.

Hints

Hint 1
The reference input is where the exponent becomes $0$ . In a form with $x-3$ in the exponent, what must the coefficient equal when $x=3$ ?
Hint 2
$f(3)$ means replace every $x$ in the given rule with $3$ . Define the full function in Desmos, then evaluate $f(3)$ rather than working through the powers separately.
Hint 3
The base tells you what multiplies the output when $x$ increases by $1$ . How could you use $f(3)$ and $f(4)$ to find that multiplier?

Step-by-step

Evaluate the reference value and multiplier
Step 1
Identify what the coefficient must show
In a form $A b^{x-3}$ , plugging in $x=3$ makes the exponent $0$ . Any positive base to the power $0$ is $1$ , so $f(3)=A$ . The problem calls $f(3)$ $k$ , which means the coefficient must equal $k$ . A matching coefficient isn't enough, though: the exponent and base must also give the same function.
Step 2
Find the coefficient
Type the given rule as $f(x)$ , then type $f(3)$ . Desmos prints $15{,}552$ , so $k=15{,}552$ . Keep each entire exponent together as you enter the function.
Step 3
Find the one-unit multiplier
Each exponent in the given function is linear in $x$ , so increasing $x$ by $1$ multiplies the output by the same factor each time. Add $f(4)/f(3)$ to find that factor. Desmos shows $1.333\ldots$ ; its fraction button gives $\frac43$ . That's the base. Reversing the ratio would give $\frac34$ instead.
Step 4
Choose the equivalent form
Use both results: $f(x)=15{,}552\left(\frac43\right)^{x-3}$ . At $x=3$ , its power is $1$ , so its coefficient shows $k$ ; each one-unit increase then multiplies the output by $\frac43$ . Choice A.

Question 194·200 Super-Hard SAT Math Questions·Advanced Math

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