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Function f is defined by f(x) = (x+6)(x+5)(x+1). Function g is defined = f(x−1) y = g(x) by g(x). The graph of in the (a,0) (b,0) (c,0) xy-plane has x-intercepts at,, and, where a, b, and c are distinct constants. What is the a+b+c value of? −15
Correct answer: B (parsed from the explanation — this source omits an answer key for this item)
Explanation
= - = +6 +5 +1 Choice B is correct. It’s given that g^xh f^x 1h. Since f^xh ^x h^x h^x h, -1 = -1+6 -1+5 -1+1 it follows that f^x h ^x h^x h^x h. Combining like terms -1 = +5 +4 = +5 +4 yields f^x h ^x h^x h^xh. Therefore, g^xh x^x h^x h. The = x-intercepts of a graph in the xy-plane are the points where y 0. The = x-coordinates of the x-intercepts of the graph of y g^xh in the xy-plane can be 0= +5 +4 found by solving the equation x^x h^x h. Applying the zero product = +5=0 +4=0 property to this equation yields three equations: x 0, x , and x . = =- =- Solving each of these equations for x yields x 0, x 5, and x 4, = 0,0 respectively. Therefore, the x-intercepts of the graph of y g^xh are ^ h, -5,0 -4,0 - - ^ h, and ^ h. It follows that the values of a, b, and c are 0, 5, and 4. + + 0+ -5 + -4 - Thus, the value of a b c is ^ h ^ h, which is equal to 9. + + = + Choice A is incorrect. This is the value of a b c if g^xh f^x 1h. Choice C is + + - = -6 -51 - incorrect. This is the value of a b c 1 if g^xh ^x h^xx h^ h. Choice D + + = -6 -5 -1 is incorrect. This is the value of a b c if f^xh ^x h^x h^x h.
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