The given polynomial function is
[tex]f(x)=-x^3+15x^2-75x+125[/tex]
It is a cubic function because the greatest power of x is 3
Since there is no value of x can make f(x) undefined, then
The domain is all real numbers
[tex]D=(-\infty,\infty)[/tex]
The range is the values of y corresponding to x, then
The range is all the real number
[tex]R=(-\infty,\infty)[/tex]
The end of behavior is
Rises to left and falls to the right
The turning point is the point that the tangent to the curve at this point = 0
Then to find it differentiate f(x) and equate it by 0 to get the x-coordinate of it
[tex]\begin{gathered} f^{\prime}(x)=-3x^2+30x-75 \\ f^{\prime}(x)=0 \\ -3x^2+30x-75=0 \\ \frac{-3x^2}{-3}+\frac{30x}{-3}-\frac{75}{-3} \\ x^2-10x+25=0 \\ (x-5)^2=0 \\ x-5=0 \\ x=5 \\ f(5)=-125+375-375-125 \\ f(5)=0 \end{gathered}[/tex]
The turning point is (5, 0)
The function f(x) is a polynomial function of 3rd degree, has a domain all real numbers and range all real numbers, its end behavior is rises at left and falls at right
