Given:
given functions are
[tex]f(x)=x^4-4x^3-2x^2-12x+9,g(x)=\sqrt{x^2-2x-3},h(x)=\frac{-x^2+1}{x^2-2x-3}[/tex]Find:
(A) we have to compare the Domain and range of the function f(x) and g(x).
(B) We have to find the relationship between the break of h(x) and zeros of f(x).
Explanation:
The domain and Range of f(x) is
[tex]\begin{gathered} Domain(f)=(-\infty,\infty) \\ Range(f)=[0,\infty) \end{gathered}[/tex]Domain and Range of g(x) is
[tex]\begin{gathered} Domain(g)=(-\infty,-1]\cup[3,\infty) \\ Range(g)=[0,\infty) \end{gathered}[/tex]Domain of h(x) is
[tex]Domain(h)=(-\infty.-1)\cup(-1,3)\cup(3,\infty)[/tex]Now zeros of the function f(x) are
[tex]\begin{gathered} x^4-4x^3-2x^2+12x+9=0 \\ (x+1)^2(x-3)^2=0 \\ x=-1,-1,3,3 \end{gathered}[/tex]Therefore, zeors of the function f(x) are -1,-1,3,3.
Now,
(A)The difference between Domain of f(x) and g(x) is of the interval (-1,3). The domain of f(x) is all Real Numbers and Domain of g(x) is all the real number except the interval (-1,3).
The Range of both f(x) and g(x) is same.
(B) The breaks in the Domain of h(x) are equal to the zeros -1, 3 of f(x).
