Question

The function f(x) = (x − 4)(x − 2) is shown. what is the range of the function?

Answer 1

looking at the vertex and the parabola opening we can see that all real numbers are equal to or greater then -1

Answer 2

`y\geq -1`

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

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