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Advanced Math Difficulty: Hard

The rational function $f$ is defined by an equation in the form $f (x) = \frac{a}{x + b}$ , where $a$ and $b$ are constants. The partial graph of $y = f (x)$ is shown. If $g (x) = f (x + 4)$ , which equation could define function $g$ ?

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Explanation

Choice C is correct. It's given that $f (x) = \frac{a}{x + b}$ and that the graph shown is a partial graph of $y = f (x)$ . Substituting $y$ for $f (x)$ in the equation $f (x) = \frac{a}{x + b}$ yields $y = \frac{a}{x + b}$ . The graph passes through the point $(- 7, - 2)$ . Substituting $negative 7$ for $x$ and $negative 2$ for $y$ in the equation $y = \frac{a}{x + b}$ yields $- 2 = \frac{a}{- 7 + b}$ . Multiplying each side of this equation by $- 7 + b$ yields $- 2 (- 7 + b) = a$ , or $14 - 2 b = a$ . The graph also passes through the point $(- 5, - 6)$ . Substituting $negative 5$ for $x$ and $negative 6$ for $y$ in the equation $y = \frac{a}{x + b}$ yields $- 6 = \frac{a}{- 5 + b}$ . Multiplying each side of this equation by $- 5 + b$ yields $- 6 (- 5 + b) = a$ , or $30 - 6 b = a$ . Substituting $14 minus 2 b$ for $a$ in this equation yields $30 - 6 b = 14 - 2 b$ . Adding $6 b$ to each side of this equation yields $30 = 14 + 4 b$ . Subtracting $14$ from each side of this equation yields $16 equals 4 b$ . Dividing each side of this equation by $4$ yields $4 equals b$ . Substituting $4$ for $b$ in the equation $14 - 2 b = a$ yields $14 - 2 (4) = a$ , or $6 equals a$ . Substituting $6$ for $a$ and $4$ for $b$ in the equation $f (x) = \frac{a}{x + b}$ yields $f (x) = \frac{6}{x + 4}$ . It's given that $g (x) = f (x + 4)$ . Substituting $x plus 4$ for $x$ in the equation $f (x) = \frac{6}{x + 4}$ yields $f (x + 4) = \frac{6}{x + 4 + 4}$ , which is equivalent to $f (x + 4) = \frac{6}{x + 8}$ . It follows that $g (x) = \frac{6}{x + 8}$ .

Choice A is incorrect. This could define function $g$ if $g (x) = f (x - 4)$ .

Choice B is incorrect. This could define function $g$ if $g (x) = f (x)$ .

Choice D is incorrect. This could define function $g$ if $g (x) = f (x) \cdot (x + 4)$ .