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Advanced Math / Nonlinear functions Difficulty: Hard

The functions $f$ and $g$ are defined by the given equations.

$f (x) = 3 + | - 2 x - x^{2} |$

$g (w) = | \frac{- w}{w - 1} | - w + 5$

If $f (- 4) = c$ , where $c$ is a constant, what is the value of $g (c)$ ?

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Explanation

The correct answer is $negative 4.9$ . The value of $f left parenthesis negative 4 right parenthesis$ is the value of $f (x)$ when $x = - 4$ . Substituting $negative 4$ for $x$ in the equation $f (x) = 3 + |- 2 x - x^{2}|$ yields $f (- 4) = 3 + |- 2 (- 4) - {(- 4)}^{2}|$ , or $f (- 4) = 3 + |- 8|$ , which is equivalent to $f (- 4) = 3 + 8$ , or $f (- 4) = 11$ . Since it's given that $f (- 4) = c$ , it follows that $c equals 11$ and the value of $g (c)$ is the value of $g left parenthesis 11 right parenthesis$ . Substituting $11$ for $w$ in the equation $g (w) = |\frac{- w}{w - 1}| - w + 5$ yields $g (11) = |\frac{- 11}{11 - 1}| - 11 + 5$ , or $g (11) = |- 1.1| - 6$ , which is equivalent to $g (11) = 1.1 - 6$ , or $g (11) = - 4.9$ . Note that -4.9 and -49/10 are examples of ways to enter a correct answer.