Below is a statement of a theorem about certain cubic equations. For this theorem,brepresents a real number. Theorem 1 if f is a cubic function of the form f(x) =x3−x+b and b >1,the the function has f has exactly one x-intercept. Theorem 2 If f and g are functions with g(x) =k·f(x), where k is a nonzero real number, then f and g have exactly the same x-intercepts. Using only these theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas? (a)f(x) =x3−x+ 11. (b)g(x) =x3+x+ 11. (c)h(x) =−x3+x−5. (d)r(x) =x4−x+ 11 (e)k(x) = 2x3−2x+ 11
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Below is a statement of a theorem about certain cubic equations. For this theorem,brepresents a real number.
Theorem 1 if f is a cubic function of the form f(x) =x3−x+b and b >1,the the function has f has exactly one x-intercept.
Theorem 2 If f and g are functions with g(x) =k·f(x), where k is a nonzero real number, then f and g have exactly the same x-intercepts.
Using only these theorems and some simple algebraic manipulations, what can be concluded about the functions given by the following formulas?
(a)f(x) =x3−x+ 11. (b)g(x) =x3+x+ 11. (c)h(x) =−x3+x−5. (d)r(x) =x4−x+ 11
(e)k(x) = 2x3−2x+ 11
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