Question

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

Answer 1

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

Answer 2

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