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

Answer:

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

Step-by-step 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 $-1$