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Nonlinear functions SAT question

Math · Advanced Math · Medium · Answer it here, explanation included.

x	f (x)
0	5
1	$five halves$
2	$five fourths$
3	$five eighths$

The table above gives the values of the function f for some values of x. Which of the following equations could define f ?

ExplanationShow

Choice D is correct. Each choice has a function with coefficient 5 and base 2, so the exponents must be analyzed. When the input value of x increases, the output value of f(x) decreases, so the exponent must be negative. An exponent of –x yields the values in the table: $first value, 5, which equals 5 times open parenthesis, 2 to the negative 0 power, close parenthesis$ , $second value, five halves, which equals 5 times open parenthesis, 2 to the negative 1 power, close parenthesis$ , $third value, five fourths, which equals 2 to the negative 2 power$ , and $fourth value, five eighths, which equals 5 times open parenthesis, 2 to the negative 3 power, close parenthesis$ .

Choices A and B are incorrect and may result from choosing equations that yield an increasing, rather than decreasing, output value of f(x) when the input value of x increases. Choice C is incorrect and may result from choosing an equation that doesn’t yield the values in the table.

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