Question 21·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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For positive constants $a$ and $b$ , the function $f$ is defined by $f(x)=ab^x$ . It is known that

\begin{aligned} f(2)+f(3)&=72 \\[1.4em] f(4)-f(3)&=108 \end{aligned}

The function $g$ is defined by $g(x)=f(2x-1)-6$ . Which choice gives an equation for $g$ ?

An exponential function with unknown constants and two conditions calls for finding both constants before transforming the rule. Use a two-condition Desmos regression to find them, then put the entire new input into the exponent. Exponent rules will turn the result into a number times a base raised to $x$ . Watch the $-1$ in the new input: it divides by the original base.

Hints

Hint 1
In $f(x)=ab^x$ , the exponent is the input. Replace $x$ with $2$ , $3$ , and $4$ to rewrite the two given conditions using the same $a$ and $b$ .
Hint 2
A regression can find unknown constants that satisfy several conditions. Put the two expressions from the left sides in one Desmos list and $[72,108]$ in another, keeping their order matched.
Hint 3
The input to $f$ in $f(2x-1)$ is all of $2x-1$ . After substituting the constants, use $3^{2x-1}=3^{2x}/3$ : the $-1$ divides by $3$ .

Step-by-step

Fit the constants, then rewrite the exponent
Step 1
Rewrite the two conditions
Since $f(k)=ab^k$ , replace each function value with its rule. The same constants must satisfy both equations:
$\begin{aligned} ab^2+ab^3&=72\\[1.4em] ab^4-ab^3&=108. \end{aligned}$
Step 2
Find both constants in Desmos
Type the matched lists below. The $\sim$ tells Desmos to fit $a$ and $b$ ; the braces enforce the given positive values:
$[ab^2+ab^3,ab^4-ab^3]\sim[72,108]\{a>0\}\{b>0\}.$
Under PARAMETERS, Desmos shows $a=2$ and $b=3$ .
Step 3
Put the new input into the rule
Inside $f$ is its input: replace $x$ by the entire $2x-1$ in the exponent. Keep the $-6$ outside:
$g(x)=ab^{2x-1}-6.$
Step 4
Substitute the fitted constants
Use the values Desmos found, $a=2$ and $b=3$ :
$g(x)=2\cdot3^{2x-1}-6.$
Step 5
Account for the minus one
Subtracting $1$ in an exponent divides by the base. It doesn't multiply by $3$ :
$g(x)=\frac{2}{3}\cdot3^{2x}-6.$
Step 6
Make the exponent $x$
Use the power-of-a-power rule, $3^{2x}=(3^2)^x$ . The $2$ in the exponent squares the base, not the coefficient:
$g(x)=\frac{2}{3}\cdot(3^2)^x-6.$
Step 7
Name the new base
Since $3^2=9$ , the equation for the transformed function is
$g(x)=\frac{2}{3}\cdot9^x-6.$
Choice D.

Question 21·25 Super-Hard SAT Advanced Math Questions·Advanced Math

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