Moving from left to right: The curve is in quadrant 3. The curve trends down sharply. As x increases, the

Question

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 $g$ ?

Accepted Answer

Choice C is correct. It's given that f x = 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 = a x + b yields y = a x + b . The graph passes through the point - 7 , - 2 . Substituting - 7 for x and - 2 for y in the equation y = a x + b yields - 2 = 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 - 5 for x and - 6 for y in the equation y = a x + b yields - 6 = a - 5 + b . Multiplying each side of this equation by - 5 + b yields - 6 - 5 + b = a , or 30 - 6 b = a . Substituting 14 - 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 = 4 b . Dividing each side of this equation by 4 yields 4 = b . Substituting 4 for b in the equation 14 - 2 b = a yields 14 - 2 4 = a , or 6 = a . Substituting 6 for a and 4 for b in the equation f x = a x + b yields f x = 6 x + 4 . It's given that g x = f x + 4 . Substituting x + 4 for x in the equation f x = 6 x + 4 yields f x + 4 = 6 x + 4 + 4 , which is equivalent to f x + 4 = 6 x + 8 . It follows that g x = 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 /mo> x + 4 .

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