Answer:
\(y\geq -1\)
All real numbers greater than or equal to \(-1\)
Step-by-step explanation:
we know that
The equation of a vertical parabola in vertex form is equal to
\(f(x)=a(x-h)^{2}+k\)
where
(h,k) is the vertex of the parabola
if \(a> 0\) ----> the parabola open upward ( vertex is a minimum)
if \(a< 0\) ----> the parabola open downward ( vertex is a maximum)
In this problem we have
\(f(x)=(x-4)(x-2)\)
Convert to vertex form
\(f(x)=(x-4)(x-2)=x^{2} -2x-4x+8\\f(x)=x^{2}-6x+8\)
Group terms that contain the same variable, and move the constant to the opposite side of the equation
\(f(x)-8=x^{2}-6x\)
Complete the square. Remember to balance the equation by adding the same constants to each side.
\(f(x)-8+9=x^{2}-6x+9\)
\(f(x)+1=x^{2}-6x+9\)
Rewrite as perfect squares
\(f(x)+1=(x-3)^{2}\)
\(f(x)=(x-3)^{2}-1\) -------> equation in vertex form
The vertex is the point \((3,-1)\)
\(a=1\)
therefore
\(a> 0\) ----> the parabola open upward ( vertex is a minimum)
The range of the function is the interval-------> [-1,∞)
\(y\geq -1\)
All real numbers greater than or equal to \(-1\)
see the attached figure to better understand the problem
