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Advanced Math / Nonlinear functions Difficulty: Hard

The function $g$ is defined by $g (x) = \frac{|x|}{a} - 14$ , where $a less than 0$ . What is the product of $g left parenthesis 15 a right parenthesis$ and $g left parenthesis 7 a right parenthesis$ ?

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Explanation

The correct answer is $609$ . It’s given that the function $g$ is defined by $g (x) = \frac{|x|}{a} - 14$ , where $a less than 0$ . Substituting $15 a$ for $x$ in function $g$ yields $g (15 a) = \frac{|15 a|}{a} - 14$ . This function can be rewritten as $g (15 a) = \frac{15 |a|}{a} - 14$ , or $g (15 a) = 15 (\frac{|a|}{a}) - 14$ . Since $a less than 0$ , it follows that $\frac{|a|}{a} = - 1$ . Substituting $negative 1$ for $\frac{|a|}{a}$ in $g (15 a) = 15 (\frac{|a|}{a}) - 14$ yields $g (15 a) = 15 (- 1) - 14$ , or $g (15 a) = - 29$ . Similarly, substituting $7 a$ for $x$ in function $g$ yields $g (7 a) = \frac{|7 a|}{a} - 14$ . This function can be rewritten as $g (7 a) = \frac{7 |a|}{a} - 14$ , or $g (7 a) = 7 (\frac{|a|}{a}) - 14$ . Since $a less than 0$ , it again follows that $\frac{|a|}{a} = - 1$ . Substituting $negative 1$ for $\frac{|a|}{a}$ in $g (7 a) = 7 (\frac{|a|}{a}) - 14$ yields $g (7 a) = 7 (- 1) - 14$ , or $g (7 a) = - 21$ . Therefore, $g (15 a) = - 29$ and $g (7 a) = - 21$ . Thus, the product of $g left parenthesis 15 a right parenthesis$ and $g left parenthesis 7 a right parenthesis$ is $(- 29) (- 21)$ , or $609$ .