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Advanced Math / Nonlinear functions Difficulty: Hard

The graph of $y = f (x)$ $y = f (x)$ is shown, where $f (x) = a b^{x} + c$ $f (x) = a b^{x} + c$ , and $a$ $a$ , $b$ $b$ , and $c$ $c$ are constants. For how many values of $x$ $x$ does $f (x) = 0$ $f left parenthesis x right parenthesis equals 0$ ?

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Explanation

Choice D is correct. Each point $(x, y)$ $(x, y)$ on the graph of $y = f (x)$ $y = f (x)$ in the xy-plane gives a value of $x$ $x$ and its corresponding value of $f (x)$ $f (x)$ , or $y$ $y$ . For any value of $x$ $x$ for which $f (x) = 0$ $f left parenthesis x right parenthesis equals 0$ , there is a corresponding point on the graph of $y = f (x)$ $y = f (x)$ with a y-coordinate of $0$ $0$ . A point with a y-coordinate of $0$ $0$ is a point where the graph intersects the x-axis. It's given that $f (x) = a b^{x} + c$ $f (x) = a b^{x} + c$ , where $a$ $a$ , $b$ $b$ , and $c$ $c$ are constants. In the xy-plane, the graph of an equation of this form will lie entirely either above or below the horizontal line defined by $y = c$ $y = c$ . The part of the graph of $y = f (x)$ $y = f (x)$ shown lies entirely below the horizontal line defined by $y = - 7$ $y = - 7$ , and thus the entire graph of $y = f (x)$ $y = f (x)$ must lie below the line defined by $y = - 7$ $y = - 7$ . It follows that the graph of $y = f (x)$ $y = f (x)$ will never intersect the x-axis. Therefore, there are zero values of x for which $f (x) = 0$ $f left parenthesis x right parenthesis equals 0$ .

Choice A is incorrect and may result from conceptual errors.

Choice B is incorrect and may result from conceptual errors.

Choice C is incorrect and may result from conceptual errors.