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Advanced Math Difficulty: Hard

$f left parenthesis x right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x$

The function $f$ is defined by the given equation. If $g (x) = f (x + 2)$ , which of the following equations defines the function $g$ ?

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Explanation

Choice B is correct. It’s given that $f left parenthesis x right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x$ and $g (x) = f (x + 2)$ . Substituting $x plus 2$ for $x$ in $f left parenthesis x right parenthesis equals 9 left parenthesis 4 right parenthesis Superscript x$ gives $f (x + 2) = 9 {(4)}^{x + 2}$ . Rewriting this equation using properties of exponents gives $f (x + 2) = 9 {(4)}^{x} {(4)}^{2}$ , which is equivalent to $f (x + 2) = 9 {(4)}^{x} (16)$ . Multiplying $9$ and $16$ in this equation gives $f (x + 2) = 144 {(4)}^{x}$ . Since $g (x) = f (x + 2)$ , $g left parenthesis x right parenthesis equals 144 left parenthesis 4 right parenthesis Superscript x$ .

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

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

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