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Advanced Math Difficulty: Hard

$f left parenthesis x right parenthesis equals 272 left parenthesis 2 right parenthesis Superscript x$

The function $f$ is defined by the given equation. If $h (x) = f (x - 4)$ , which of the following equations defines function $h$ ?

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Explanation

Choice A is correct. It's given that $f left parenthesis x right parenthesis equals 272 left parenthesis 2 right parenthesis Superscript x$ and $h (x) = f (x - 4)$ . Substituting $x minus 4$ for $x$ in $f left parenthesis x right parenthesis equals 272 left parenthesis 2 right parenthesis Superscript x$ yields $f (x - 4) = 272 {(2)}^{x - 4}$ . Substituting $h (x)$ for $f left parenthesis x minus 4 right parenthesis$ in this equation yields $h (x) = 272 {(2)}^{x - 4}$ . Using the properties of exponents, the function $h (x) = 272 (2)^{x - 4}$ can be rewritten as $h (x) = \frac{272 {(2)}^{x}}{2^{4}}$ , which is equivalent to $h (x) = \frac{272 {(2)}^{x}}{16}$ , or $h left parenthesis x right parenthesis equals 17 left parenthesis 2 right parenthesis Superscript x$ . Therefore, of the given choices, an equation that defines function $h$ is $h left parenthesis x right parenthesis equals 17 left parenthesis 2 right parenthesis Superscript x$ .

Choice B is incorrect. This equation defines function $h$ if $h (x) = f (x - 2)$ , not $h (x) = f (x - 4)$ .

Choice C is incorrect. This equation defines function $h$ if $h left parenthesis x right parenthesis equals f left parenthesis 4 x right parenthesis$ , not $h (x) = f (x - 4)$ .

Choice D is incorrect and may result from conceptual or calculation errors.