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Advanced Math / Nonlinear functions Difficulty: Hard

The function $f$ is defined by $f (x) = a \sqrt{x + b}$ , where $a$ and $b$ are constants. In the xy-plane, the graph of $y = f (x)$ passes through the point $left parenthesis negative 24 comma 0 right parenthesis$ , and $f left parenthesis 24 right parenthesis less than 0$ . Which of the following must be true?

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Explanation

Choice D is correct. It's given that $f left parenthesis 24 right parenthesis less than 0$ . Substituting $24$ for $f (x)$ in the equation $f (x) = a \sqrt{x + b}$ yields $f (24) = a \sqrt{24 + b}$ . Therefore, $a StartRoot 24 plus b EndRoot less than 0$ . Since $StartRoot 24 plus b EndRoot$ can't be negative, it follows that $a less than 0$ . It's also given that the graph of $y = f (x)$ passes through the point $left parenthesis negative 24 comma 0 right parenthesis$ . It follows that when $x = - 24$ , $f left parenthesis x right parenthesis equals 0$ . Substituting $negative 24$ for $x$ and $0$ for $f (x)$ in the equation $f (x) = a \sqrt{x + b}$ yields $0 = a \sqrt{- 24 + b}$ . By the zero product property, either $a equals 0$ or $\sqrt{- 24 + b} = 0$ . Since $a less than 0$ , it follows that $\sqrt{24 + b} = 0$ . Squaring both sides of this equation yields $- 24 + b = 0$ . Adding $24$ to both sides of this equation yields $b equals 24$ . Since $a less than 0$ and $b$ is $24$ , it follows that $a < b$ must be true.

Choice A is incorrect. The value of $f left parenthesis 0 right parenthesis$ is $a \sqrt{b}$ , which must be negative.

Choice B is incorrect. The value of $f left parenthesis 0 right parenthesis$ is $a \sqrt{b}$ , which could be $negative 24$ , but doesn't have to be.

Choice C is incorrect and may result from conceptual or calculation errors.