Answer:
All real numbers greater than or equal to [tex]-1[/tex]
Step-by-step explanation:
we have
[tex]f(x) = (x -4)(x- 2)[/tex]
Multiply
[tex]f(x)=x^{2}-2x-4x+8[/tex]
[tex]f(x)=x^{2}-6x+8[/tex]
This is a vertical parabola open upward
Convert into vertex form
The equation of a vertical parabola into vertex form is equal to
[tex]y=a(x-h)^{2}+k[/tex]
where (h,k) is the vertex
Group terms that contain the same variable, and move the constant to the opposite side of the equation
[tex]f(x)-8=x^{2}-6x[/tex]
Complete the square. Remember to balance the equation by adding the same constants to each side
[tex]f(x)-8+9=x^{2}-6x+9[/tex]
[tex]f(x)+1=x^{2}-6x+9[/tex]
Rewrite as perfect squares
[tex]f(x)+1=(x-3)^{2}[/tex]
[tex]f(x)=(x-3)^{2}-1[/tex]
The vertex is the point [tex](3,-1)[/tex] ------> is a minimum (open upward)
therefore
The range is the interval -------> [-1,∞)
All real numbers greater than or equal to [tex]-1[/tex]
