The function f(x) = (x - 4)(x - 2) is shown. What is the range of the function?
Solution:
The equation of a vertical parabola in vertex form is
f(x) = a(x - h)2 + k
Where (h, k) is the vertex
The parabola opens upward if a > 0 (vertex is minimum)
The parabola opens downward if a < 0 (vertex is maximum)
The function given is
f(x) = (x - 4) (x - 2)
Let us convert it to vertex form
f(x) = x2 - 2x - 4x + 8
f(x) = x2 - 6x + 8
We can write it as
f(x) - 8 = x2 - 6x
Let us add 9 on both sides
f(x) - 8 + 9 = x2 - 6x + 9
f(x) + 1 = x2 - 6x + 9
By rewriting it as perfect squares
f(x) + 1 = (x - 3)2
f(x) = (x - 3)2 - 1
So the vertex is the point (3, -1)
a = 1
Here a >0
The range of the function is [-1, ∞)
y ≥ -1
Which is all real numbers which are greater than or equal to -1
Therefore, the range of the function is all real numbers which are greater than or equal to -1.
The function f(x) = (x - 4)(x - 2) is shown. What is the range of the function?
Summary:
The function f(x) = (x - 4)(x - 2) is shown. The range of the function is all real numbers which are greater than or equal to -1.