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

Explanation:

Step1: Analyze the parabola's direction

The function \( f(x)=(x - 4)(x - 2) \) is a quadratic function. Expanding it: \( f(x)=x^{2}-6x + 8 \). The coefficient of \( x^{2} \) is positive (1), so the parabola opens upwards.

Step2: Find the vertex (minimum point)

For a quadratic function \( y = ax^{2}+bx + c \), the x - coordinate of the vertex is \( x=-\frac{b}{2a} \). Here, \( a = 1 \), \( b=-6 \), so \( x =-\frac{-6}{2\times1}=3 \).
Substitute \( x = 3 \) into the function: \( f(3)=(3 - 4)(3 - 2)=(-1)\times1=-1 \). So the vertex is at \( (3,-1) \).
Since the parabola opens upwards, the minimum value of the function is \( y=-1 \), and the range is all real numbers greater than or equal to - 1.

Answer:

all real numbers greater than or equal to -1 (the option: all real numbers greater than or equal to -1)