The function f is defined by f x = a x + b , where a and b are constants.

Question

The function f is defined by f x = a x + b , where a and b are constants. In the xy -plane, the graph of y = f x passes through the point - 24 , 0 , and f 24 &#60; 0 . Which of the following must be true?

Accepted Answer

Choice D is correct. It's given that f 24 &#60; 0 . Substituting 24 for f x in the equation f x = a x + b yields f 24 = a 24 + b . Therefore, a 24 + b &#60; 0 . Since 24 + b can't be negative, it follows that a &#60; 0 . It's also given that the graph of y = f x passes through the point - 24 , 0 . It follows that when x = - 24 , f x = 0 . Substituting - 24 for x and 0 for f x in the equation f x = a x + b yields 0 = a - 24 + b . By the zero product property, either a = 0 or - 24 + b = 0 . Since a &#60; 0 , it follows that 24 + b = 0 . Squaring both sides of this equation yields - 24 + b = 0 . Adding 24 to both sides of this equation yields b = 24 . Since a &#60; 0 and b is 24 , it follows that a &#60; b must be true. Choice A is incorrect. The value of f 0 is a b , which must be negative. Choice B is incorrect. The value of f 0 is a b , which could be - 24 , but doesn't have to be. Choice C is incorrect and may result from conceptual or calculation errors.

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