Question 17·200 Super-Hard SAT Math Questions·Advanced Math

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Let

f(x)=\frac{4^{x+1}\cdot 9^{2x-3}}{6^{3x-5}}.

For this exponential function $f$ , let $k=f(-1)$ . Which choice rewrites $f(x)$ so that $k$ appears as the coefficient (the number multiplying the exponential term)?

In an exponential function, the coefficient is the number multiplying the power. That number equals the function's output where the exponent is zero. If the question names an output $k=f(t)$ , make the exponent zero at $t$ , evaluate $k$ in Desmos, and find the multiplier for each increase of $1$ in $x$ . Equivalent forms can show different numbers as their coefficients.

Hints

Hint 1
The coefficient is the number in front of an exponential power. It equals the function's output when that power is $1$ . What exponent makes a positive base raised to it equal $1$ ?
Hint 2
To find $k=f(-1)$ , define the given function in Desmos and type $f(-1)$ on the next line. Use the fraction button to compare the exact value with the numbers in front of the choices.
Hint 3
Increasing $x$ by $1$ multiplies the numerator's powers by $4$ and $9^2$ , while the denominator gains a factor of $6^3$ . The overall multiplier divides the numerator's new factors by the denominator's.

Step-by-step

Find the coefficient at the zero exponent
Step 1
Make the exponent zero at the given input
The coefficient is the number multiplying the exponential power. The problem sets $k=f(-1)$ , so that power must equal $1$ at $x=-1$ . The zero-exponent rule says $b^0=1$ for a positive base. Since $x+1=0$ at $x=-1$ , an exponent of $x+1$ makes the coefficient equal $f(-1)$ .
Step 2
Evaluate the coefficient
Type the given rule as $f(x)=\frac{4^{x+1}\cdot9^{2x-3}}{6^{3x-5}}$ , then type $f(-1)$ . Desmos displays $28.444\ldots$ ; its fraction button shows $\frac{256}{9}$ . So $k=\frac{256}{9}$ . Keep the negative input inside $f(-1)$ rather than dropping its sign.
Step 3
Find the multiplier for one increase in x
When $x$ increases by $1$ , the numerator gains factors of $4$ and $9^2$ ; the denominator gains $6^3$ . Type $\frac{4\cdot9^2}{6^3}$ on a new line. Desmos displays $1.5$ , or $\frac32$ with the fraction button. So the per-unit multiplier is $\frac32$ .
Step 4
Put k in front
Use $k=\frac{256}{9}$ , the multiplier $\frac32$ , and the exponent $x+1$ : $f(x)=\frac{256}{9}\left(\frac32\right)^{x+1}$ . At $x=-1$ , the power is $1$ , so the number in front is exactly $k$ . Choice C.

Question 17·200 Super-Hard SAT Math Questions·Advanced Math

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