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Advanced Math / Nonlinear functions Difficulty: Hard

For the function $f$ , $f left parenthesis 0 right parenthesis equals 86$ , and for each increase in $x$ by $1$ , the value of $f (x)$ decreases by $80 percent sign$ . What is the value of $f left parenthesis 2 right parenthesis$ ?

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Explanation

The correct answer is $3.44$ . It’s given that $f left parenthesis 0 right parenthesis equals 86$ and that for each increase in $x$ by $1$ , the value of $f (x)$ decreases by $80 percent sign$ . Because the output of the function decreases by a constant percentage for each $1$ -unit increase in the value of $x$ , this relationship can be represented by an exponential function of the form $f (x) = a {(b)}^{x}$ , where $a$ represents the initial value of the function and $b$ represents the rate of decay,
expressed as a decimal. Because $f left parenthesis 0 right parenthesis equals 86$ , the value of $a$ must be $86$ . Because the value of $f (x)$ decreases by $80 percent sign$ for each $1$ -unit increase in $x$ , the value of $b$ must be $left parenthesis 1 minus 0 period 80 right parenthesis$ , or $0 period 2$ . Therefore, the function $f$ can be defined by $f left parenthesis x right parenthesis equals 86 left parenthesis 0 period 2 right parenthesis Superscript x$ . Substituting $2$ for $x$ in this function yields $f left parenthesis 2 right parenthesis equals 86 left parenthesis 0 period 2 right parenthesis squared$ , which is equivalent to $f left parenthesis 2 right parenthesis equals 86 left parenthesis 0 period 04 right parenthesis$ , or $f left parenthesis 2 right parenthesis equals 3 period 44$ . Either $3.44$ or $86 divided by 25$ may be entered as the correct answer.

Alternate approach: It’s given that $f left parenthesis 0 right parenthesis equals 86$ and that for each increase in $x$ by $1$ , the value of $f (x)$ decreases by $80 percent sign$ . Therefore, when $x equals 1$ , the value of $f (x)$ is $left parenthesis 100 minus 80 right parenthesis percent sign$ , or $20 percent sign$ , of $86$ , which can be expressed as $left parenthesis 0 period 20 right parenthesis left parenthesis 86 right parenthesis$ . Since $left parenthesis 0 period 20 right parenthesis left parenthesis 86 right parenthesis equals 17 period 2$ , the value of $f left parenthesis 1 right parenthesis$ is $17.2$ . Similarly, when $x equals 2$ , the value of $f (x)$ is $20 percent sign$ of $17.2$ , which can be expressed as $left parenthesis 0 period 20 right parenthesis left parenthesis 17 period 2 right parenthesis$ . Since $left parenthesis 0 period 20 right parenthesis left parenthesis 17 period 2 right parenthesis equals 3 period 44$ , the value of $f left parenthesis 2 right parenthesis$ is $3.44$ . Either $3.44$ or $86 divided by 25$ may be entered as the correct answer.