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Advanced Math / Nonlinear functions Difficulty: Medium

$x$ $x$	$g (x)$ $g (x)$
$- 1$ $negative 1$	$25$ $25$
$0$ $0$	$1$ $1$
$1$ $1$	$\frac{1}{25}$
$2$	$StartFraction 1 Over 625 EndFraction$

For the exponential function $g$ , the table shows four values of $x$ and their corresponding values of $g (x)$ . Which equation defines $g$ ?

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Explanation

Choice D is correct. It's given that function $g$ is exponential. Therefore, an equation defining $g$ can be written in the form $g (x) = a {(b)}^{x}$ , where $a$ and $b$ are constants. The table shows that when $x equals 0$ , $g left parenthesis x right parenthesis equals 1$ . Substituting $0$ for $x$ and $1$ for $g (x)$ in the equation $g (x) = a {(b)}^{x}$ yields $1 equals a left parenthesis b right parenthesis Superscript 0$ , which is equivalent to $1 equals a$ . Substituting $1$ for $a$ in the equation $g (x) = a {(b)}^{x}$ yields $g (x) = {(b)}^{x}$ . The table also shows that when $x equals 1$ , $g (x) = \frac{1}{25}$ . Substituting $1$ for $x$ and $\frac{1}{25}$ for $g (x)$ in the equation $g (x) = {(b)}^{x}$ yields $one twenty fifth equals left parenthesis b right parenthesis Superscript 1$ , which is equivalent to $\frac{1}{25} = b$ . Substituting $\frac{1}{25}$ for $b$ in the equation $g (x) = {(b)}^{x}$ yields $g (x) = {(\frac{1}{25})}^{x}$ .

Choice A is incorrect. For this function, $g left parenthesis 1 right parenthesis$ is equal to $negative 25$ , not $\frac{1}{25}$ .

Choice B is incorrect. For this function, $g left parenthesis 1 right parenthesis$ is equal to $- \frac{1}{25}$ , not $\frac{1}{25}$ .

Choice C is incorrect. For this function, $g left parenthesis 1 right parenthesis$ is equal to $25$ , not $\frac{1}{25}$ .