Question

The function f(x) = (x − 4)(x − 2) is shown.What is the range of the function? all real numbers less than or equal to 3, all real numbers less than or equal to −1, all real numbers greater than or equal to 3, all real numbers greater than or equal to −1

Answer 1

The range is [-1 , ∞ )

So, all real numbers greater than or equal to -1.

Answer 2

All real numbers greater than or equal to
`-1`

Explanation:

we have

`f(x) = (x -4)(x- 2)`

Multiply

`f(x)=x^(2)-2x-4x+8`

`f(x)=x^(2)-6x+8`

This is a vertical parabola open upward

Convert into vertex form

The equation of a vertical parabola into vertex form is equal to

`y=a(x-h)^(2)+k`

where (h,k) is the vertex

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`

The vertex is the point
`(3,-1)` ------> is a minimum (open upward)

therefore

The range is the interval -------> [-1,∞)

All real numbers greater than or equal to
`-1`