Super-Hard SAT Math Question 45 | Free SAT Prep | Aniko

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Question 45·200 Super-Hard SAT Math Questions·Advanced Math

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The functions $f$ and $g$ are defined by the given equations, where $x\ge 0$ . Which of the following equations is written so that the input value that produces an output of $64$ can be read directly from the exponent, without algebraic rewriting?

I. $f(x)=64\left(\frac{5}{4}\right)^2\left(\frac{4}{5}\right)^x$

II. $g(x)=64\left(\frac{4}{5}\right)^{x-2}$

For an exponential expression written as $A\cdot b^{x-c}$ , recognize that the exponent becomes $0$ at $x=c$ , making the exponential factor equal to $1$ . Therefore, the output at that input is $A$ . When a question asks what an equation displays directly, focus on whether the relevant shift or coefficient is already visible in the equation rather than rewriting an equivalent form.

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Use a zero exponent

Recall that for any nonzero number $a$ , $a^0=1$ .

Look for a shifted exponent

An exponent of the form $x-c$ becomes $0$ when $x=c$ .

Compare the equation forms

Determine which equation has an exponent that directly reveals when its exponential factor equals $1$ .

Desmos Guide

Enter the equations

Enter the following expressions in Desmos:

$y_1=64\left(\frac{5}{4}\right)^2\left(\frac{4}{5}\right)^x$
$y_2=64\left(\frac{4}{5}\right)^{x-2}$

Check the output at the shifted input

Use a table or evaluate both expressions at $x=2$ . Each has output $64$ , confirming that the equations define the same function.

Compare the displayed forms

Observe that only the second equation contains the exponent $x-2$ . This form directly shows that the exponential factor equals $1$ at $x=2$ , whereas the first equation requires simplification or solving to identify that input.

Step-by-step Explanation

Identify the useful exponent form

An expression of the form $a^{x-c}$ has an exponent of $0$ when $x=c$ . At that input, $a^{x-c}=a^0=1$ .

Apply this idea to equation II

In II, the exponential factor is $\left(\frac{4}{5}\right)^{x-2}$ . When $x=2$ , its exponent is $0$ , so

g(2)=64\left(\frac{4}{5}\right)^0=64.

Thus, the shifted exponent $x-2$ directly identifies the input $2$ that produces an output of $64$ .

Contrast equation I

Equation I is written with exponent $x$ , not $x-2$ . Its coefficient is

64\left(\frac{5}{4}\right)^2=100,

so finding the input that produces an output of $64$ would require algebraic rewriting or solving an equation. The input cannot be read directly from its exponent.

Select the equation

Only equation II directly displays the needed input in the exponent.

Correct answer: II only

Strategy

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Desmos Guide

Step-by-step Explanation

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Step-by-step Explanation