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Advanced Math / Nonlinear functions Difficulty: Medium

For the exponential function $f$ , the value of $f left parenthesis 0 right parenthesis$ is $c$ , where $c$ is a constant. Of the following equations that define the function $f$ , which equation shows the value of $c$ as the coefficient or the base?

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Explanation

Choice B is correct. Each of the given choices is an equation of the form $f (x) = a {(b)}^{x - k}$ , where $a$ , $b$ , and $k$ are constants. For an equation of this form, the coefficient, $a$ , is equal to the value of the function when the exponent is equal to $0$ , or when $x = k$ . It follows that in the equation $f left parenthesis x right parenthesis equals 33 left parenthesis 1.5 right parenthesis Superscript x$ , the coefficient, $33$ , is equal to the value of $f left parenthesis 0 right parenthesis$ . Substituting $0$ for $x$ in this equation yields $f left parenthesis 0 right parenthesis equals 33 left parenthesis 1.5 right parenthesis Superscript 0$ , which is equivalent to $f left parenthesis 0 right parenthesis equals 33 left parenthesis 1 right parenthesis$ , or $f left parenthesis 0 right parenthesis equals 33$ . Thus, the value of $c$ is $33$ and the equation $f left parenthesis x right parenthesis equals 33 left parenthesis 1.5 right parenthesis Superscript x$ shows the value of $c$ as the coefficient.

Choice A is incorrect. This equation shows the value of $f left parenthesis negative 1 right parenthesis$ , not $f left parenthesis 0 right parenthesis$ , as the coefficient.

Choice C is incorrect. This equation shows the value of $f left parenthesis 1 right parenthesis$ , not $f left parenthesis 0 right parenthesis$ , as the coefficient.

Choice D is incorrect. This equation shows the value of $f left parenthesis 2 right parenthesis$ , not $f left parenthesis 0 right parenthesis$ , as the coefficient.