Certainly! Let's analyze the function [tex]\( f(x) = x^2 - 4x + 2 \)[/tex] step by step to determine its range and end behavior.
1. Determining the Range in Set Notation:
- The function [tex]\( f(x) = x^2 - 4x + 2 \)[/tex] is a quadratic function that opens upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive).
- To find the minimum value of the function, we find the vertex. The vertex of a parabola [tex]\( ax^2 + bx + c \)[/tex] occurs at [tex]\( x = -\frac{b}{2a} \)[/tex].
- For [tex]\( f(x) = x^2 - 4x + 2 \)[/tex], [tex]\( a = 1 \)[/tex] and [tex]\( b = -4 \)[/tex], so [tex]\( x = -\frac{-4}{2 \times 1} = 2 \)[/tex].
- Plugging [tex]\( x = 2 \)[/tex] back into the function gives [tex]\( f(2) = (2)^2 - 4(2) + 2 = 4 - 8 + 2 = -2 \)[/tex].
- Since the parabola opens upwards, the minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-2\)[/tex], and [tex]\( f(x) \)[/tex] can take any value greater than or equal to [tex]\(-2\)[/tex].
In set notation, the range is:
[tex]\[
\{ y \mid y \geq -2 \}
\][/tex]
2. Determining the Range in Interval Notation:
- From the previous analysis, we determined that the minimum value of [tex]\( f(x) \)[/tex] is [tex]\(-2\)[/tex], and the function takes all values greater than or equal to [tex]\(-2\)[/tex].
In interval notation, the range is:
[tex]\[
[-2, \infty)
\][/tex]
3. Describing the End Behavior of the Function:
- To describe the end behavior, we analyze what happens to [tex]\( f(x) \)[/tex] as [tex]\( x \)[/tex] approaches positive and negative infinity.
As [tex]\( x \to \infty \)[/tex]:
- The term [tex]\( x^2 \)[/tex] dominates, so [tex]\( x^2 - 4x + 2 \)[/tex] increases without bound.
- Therefore, [tex]\( f(x) \to \infty \)[/tex].
As [tex]\( x \to -\infty \)[/tex]:
- Similarly, the term [tex]\( x^2 \)[/tex] dominates, and even though [tex]\( -4x \)[/tex] is negative, [tex]\( x^2 \)[/tex] grows faster, causing [tex]\( f(x) \)[/tex] to increase without bound.
- Therefore, [tex]\( f(x) \to \infty \)[/tex].
The end behavior of the function can be described as:
[tex]\[
f(x) \to \infty \text{ as } x \to \infty
\][/tex]
and
[tex]\[
f(x) \to \infty \text{ as } x \to -\infty
\][/tex]
In summary:
- The range of the function in set notation is [tex]\( \{ y \mid y \geq -2 \} \)[/tex].
- The range in interval notation is [tex]\( [-2, \infty) \)[/tex].
- The end behavior is such that [tex]\( f(x) \to \infty \)[/tex] as [tex]\( x \to \infty \)[/tex] and [tex]\( f(x) \to \infty \)[/tex] as [tex]\( x \to -\infty \)[/tex].
