+635 The function $f(x)$ is defined only on domain $[-1,2]$, and is defined on this domain by the formula
$$f(x) = 2x^2-8x+1.$$
What is the range of $f(x)$? Express your answer as an interval or as a union of intervals.
+2653 We can find the range of the function f(x)=2x2−8x+1 by completing the square and analyzing its extreme points.
Completing the Square:
We can rewrite the function as:
f(x)=2(x2−4x)+1
To complete the square, we take half of the coefficient of our x term, square it, and add it to both sides of the equation:
Half of the coefficient of our x term is -4 / 2 = -2.
Squaring -2 gives us 4.
Therefore:
f(x)=2(x2−4x+4)+1−8
f(x)=2(x−2)2−7
Finding the Vertex and Analyzing Extremes:
The expression inside the parentheses is squared, so it's always non-negative (0 or greater). Multiplying by 2 doesn't change this property. Therefore, the minimum value of f(x) is achieved when the squared term is 0, which happens when x=2. In this case, f(2)=2(2−2)2−7=−7.
Since the coefficient of the squared term (2) is positive, the parabola opens upwards. This means the function has a minimum point at x=2 and increases as we move away from it (either to the left or right on the interval).
Range of the Function:
The function is defined only on the interval [−1,2]. We saw that the minimum value within this interval is f(2)=−7. As x approaches either end of the interval, the function increases without bound because the squared term keeps getting larger. Therefore, the range of f(x) is the single interval:
(−7,∞)
