The exponential function is defined by The value of is , where is a constant. Which choice is an equivalent form of that shows the value of as the coefficient or the base?

Solution available with step-by-step explanation

Super-Hard SAT Math Question 104 | Free SAT Prep | Aniko

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

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The exponential function $f$ is defined by Pr⁠oрertу оf Aniкo‌.аi

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?

First simplify the exponential expression by rewriting everything with the same prime bases (for example, $12^x=3^x\cdot 4^x$ ) and using exponent rules to combine terms. Once you have $f(x)=A\cdot b^x$ , compute $k=f(2)=A\cdot b^2$ , then rewrite as $f(x)=k\cdot b^{x-2}$ to make $k$ appear explicitly. Аnik‌ο -‍ F‍rее SА⁠Т Р‌reр

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Rewrite $12^x$

Use $12^x=(3\cdot 4)^x=3^x\cdot 4^x$ so you can combine the powers of $3$ . anікo.aі/ѕа⁠t

Turn division into subtraction of exponents

Write $\dfrac{1}{3^{2x-1}}=3^{1-2x}$ and then combine it with the $3^x$ coming from $12^x$ .

Use the shift identity

If you get $f(x)=A\cdot b^x$ , then $f(x)=f(2)\cdot b^{x-2}$ . Look for the option that has coefficient $f(2)$ and exponent $x-2$ .

Desmos Guide

Graph the given function

Enter y=(48*12^x)/(3^(2x-1)).

Find $k$ from a table

Add a table for the function and evaluate at x=2 to read off $k$ . Conte‌nt by ‍Аnіkο.аі

Check each choice and confirm it displays $k$

Enter each answer choice (for example, y1=256*(4/3)^(x-2)). The correct choice will match the original graph for all $x$ and have coefficient or base equal to the table value $k$ .

Step-by-step Explanation

Rewrite using common bases

Start by rewriting $12^x$ and the denominator using base factors of $3$ and $4$ :

\begin{aligned} f(x)&=\frac{48\cdot 12^x}{3^{2x-1}} =48\cdot 12^x\cdot 3^{1-2x} \\ &=48\cdot (3\cdot 4)^x\cdot 3^{1-2x} =48\cdot 3^x\cdot 4^x\cdot 3^{1-2x}. \end{aligned}

Combine like bases and compute $k$

Combine the powers of $3$ and simplify: A‍nіko.аi - ЅАТ Prep

\begin{aligned} f(x)&=48\cdot 4^x\cdot 3^{x+1-2x} =48\cdot 4^x\cdot 3^{1-x} \\ &=48\cdot 3\cdot \frac{4^x}{3^x} =144\left(\frac{4}{3}\right)^x. \end{aligned}

Now evaluate at $x=2$ :

k=f(2)=144\left(\frac{4}{3}\right)^2=144\cdot \frac{16}{9}=256.

Write in the shifted form that displays $k$

Using base $\tfrac{4}{3}$ , rewrite

144\left(\frac{4}{3}\right)^x=\left[144\left(\frac{4}{3}\right)^2\right]\left(\frac{4}{3}\right)^{x-2}=k\left(\frac{4}{3}\right)^{x-2}.

Since $k=256$ , the matching choice is

\boxed{f(x)=256\left(\frac{4}{3}\right)^{x-2}}.

Strategy

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Desmos Guide

Step-by-step Explanation

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Strategy

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Desmos Guide

Step-by-step Explanation