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Advanced Math / Nonlinear functions Difficulty: Hard

The function $f$ is defined by $f (x) = a ({2.2}^{x} + {2.2}^{b})$ , where $a$ and $b$ are integer constants and $0 less than a less than b$ . The functions $g$ and $h$ are equivalent to function $f$ , where $k$ and $m$ are constants. Which of the following equations displays the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane as a constant or coefficient?

$g (x) = a ({2.2}^{x} + k)$
$h (x) = a {(2.2)}^{x} + m$

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Explanation

Choice D is correct. A y-intercept of a graph in the xy-plane is a point where the graph intersects the y-axis, or a point where $x equals 0$ . Substituting $0$ for $x$ in the equation defining function $f$ yields $f (0) = a ({2.2}^{0} + {2.2}^{b})$ , or $f (0) = a (1 + {2.2}^{b})$ . So, the y-coordinate of the y-intercept of the graph is $a left parenthesis 1 plus 2.2 Superscript b Baseline right parenthesis$ , or equivalently, $a plus a left parenthesis 2.2 right parenthesis Superscript b$ . It's given that function $g$ is equivalent to function $f$ , where $0 less than a less than b$ . It follows that $k equals 2.2 Superscript b$ . Since $a left parenthesis 2.2 right parenthesis Superscript b$ can't be equal to $0$ , the coefficient $a$ can't be equal to $a plus a left parenthesis 2.2 right parenthesis Superscript b$ . Since $0 less than a$ , the constant $k$ , which is equal to $2.2 Superscript b$ , can't be equal to $a plus a left parenthesis 2.2 right parenthesis Superscript b$ . Therefore, function $g$ doesn't display the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane as a constant or coefficient. It's also given that function $h$ is equivalent to function $f$ , where $0 less than a less than b$ . The equation defining $f$ can be rewritten as $f (x) = a {(2.2)}^{x} + a {(2.2)}^{b}$ . It follows that $m equals a left parenthesis 2.2 right parenthesis Superscript b$ . Since $a left parenthesis 2.2 right parenthesis Superscript b$ can't be equal to $0$ , the coefficient $a$ can't be equal to $a plus a left parenthesis 2.2 right parenthesis Superscript b$ . Since $0 less than a$ , the constant $m$ , which is equal to $a left parenthesis 2.2 right parenthesis Superscript b$ , can't be equal to $a plus a left parenthesis 2.2 right parenthesis Superscript b$ . Therefore, function $h$ doesn't display the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane as a constant or coefficient. Thus, neither function $g$ nor function $h$ displays the y-coordinate of the y-intercept of the graph of $y = f (x)$ in the xy-plane as a constant or coefficient.

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

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

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