The table below gives selected values of the function . | | | | | | |-----|-----|-----|-----|-----| | | | | | | Which of the following equations could define ?

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SAT Nonlinear Functions Question 23 | Free SAT Prep | Aniko

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Question 23·Medium·Nonlinear Functions

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The table below gives selected values of the function $f$ .

$x$	$0$	$1$	$2$	$3$
$f(x)$	$5$	$\dfrac{5}{2}$	$\dfrac{5}{4}$	$\dfrac{5}{8}$

Which of the following equations could define $f$ ?

For function-from-table questions, first decide if the pattern is linear (constant difference) or exponential (constant ratio) by checking how the outputs change as x increases by 1. Once you see a constant ratio, write the function in the form $a\cdot b^x$ using the starting value at $x=0$ for $a$ and the common ratio for $b$ . Then, if needed, rewrite your expression to match the form of the answer choices (for example, using negative exponents), and quickly check by plugging in one or two x-values to confirm which option matches the table exactly.

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Check how the outputs change

Look at $f(0), f(1), f(2), f(3)$ . Are you adding or subtracting the same number each time, or are you multiplying by the same number each time?

Find the common ratio

Compute $\dfrac{f(1)}{f(0)}$ , $\dfrac{f(2)}{f(1)}$ , and $\dfrac{f(3)}{f(2)}$ . What number do you keep multiplying by as $x$ increases by 1?

Build the exponential function

Once you know the starting value $f(0)$ and the factor you multiply by each time $x$ increases by 1, think of a function of the form $a\cdot b^x$ . What are $a$ and $b$ here?

Match your function to a choice

After you write your function using $\left(\tfrac{1}{2}\right)^x$ , rewrite it using a power of 2 so you can compare it directly with the answer choices.

Desmos Guide

Enter each candidate function

In Desmos, type the four options as separate functions: y = 5*2^(x+1), y = 5*2^x, y = 5*2^(-(x+1)), and y = 5*2^(-x).

Create a small x-table for comparison

For each function, click the gear icon next to it and select "Table". In each table, enter the x-values 0, 1, 2, and 3.

Match the y-values to the given table

Compare the y-values in each Desmos table with the problem’s values $5, \dfrac{5}{2}, \dfrac{5}{4}, \dfrac{5}{8}$ at $x=0,1,2,3$ . The function whose y-values exactly match all four values is the correct choice.

Step-by-step Explanation

Look for a pattern in the table values

Write the $f(x)$ values in order: $5,\; \dfrac{5}{2},\; \dfrac{5}{4},\; \dfrac{5}{8}$ .

Now compare consecutive terms:

From $5$ to $\dfrac{5}{2}$ : you multiply by $\dfrac{1}{2}$ .
From $\dfrac{5}{2}$ to $\dfrac{5}{4}$ : you multiply by $\dfrac{1}{2}$ again.
From $\dfrac{5}{4}$ to $\dfrac{5}{8}$ : you again multiply by $\dfrac{1}{2}$ .

So every time $x$ increases by 1, $f(x)$ is multiplied by $\dfrac{1}{2}$ . This is a constant ratio, which means an exponential function of the form $a\cdot b^x$ with $b=\dfrac{1}{2}$ . The starting value (when $x=0$ ) is $5$ , so $a=5$ .

Write an exponential equation from the pattern

Using the initial value and the common ratio, we can write

Initial value (when $x=0$ ): $f(0)=5$ .
Each step in $x$ multiplies the output by $\dfrac{1}{2}$ .

So the function can be written as

f(x) = 5\left(\frac{1}{2}\right)^x.

This matches the pattern in the table (you can quickly check: $x=0$ gives $5$ , $x=1$ gives $5\cdot \tfrac{1}{2}=\tfrac{5}{2}$ , etc.).

Rewrite the function using powers of 2

The answer choices all use expressions of the form $2^{\text{(something)}}$ , so rewrite $\left(\tfrac{1}{2}\right)^x$ using a power of 2:

Note that $\dfrac{1}{2} = 2^{-1}$ .
Therefore,

\left(\frac{1}{2}\right)^x = (2^{-1})^x = 2^{-x}.

Substitute this into the equation from the previous step:

f(x) = 5\left(\frac{1}{2}\right)^x = 5(2^{-x}).

This matches choice D, so the correct equation is $f(x)=5(2^{-x})$ .

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