Math Problem Statement

Find the range of the function f(x) = x^2 - 2x + 5 for x in the interval [-2, 5].

Solution

To find the range of the function f(x)=x22x+5f(x) = x^2 - 2x + 5 for x[2,5]x \in [-2, 5], let's analyze the function step-by-step.

Step 1: Rewrite the Function in Vertex Form

The function f(x)=x22x+5f(x) = x^2 - 2x + 5 is a quadratic function, and it can be rewritten in vertex form to make it easier to analyze the minimum or maximum values.

  1. Rewrite f(x)f(x) by completing the square: f(x)=x22x+5=(x1)2+4f(x) = x^2 - 2x + 5 = (x - 1)^2 + 4

    This shows that the function can be expressed as f(x)=(x1)2+4f(x) = (x - 1)^2 + 4, where the vertex is at (1,4)(1, 4).

Step 2: Determine the Minimum Value

Since f(x)=(x1)2+4f(x) = (x - 1)^2 + 4 is a parabola that opens upwards, its minimum value occurs at the vertex x=1x = 1, where: f(1)=(11)2+4=4f(1) = (1 - 1)^2 + 4 = 4

Step 3: Evaluate the Function at the Endpoints

Next, evaluate f(x)f(x) at the endpoints of the interval x[2,5]x \in [-2, 5]:

  • For x=2x = -2: f(2)=(2)22(2)+5=4+4+5=13f(-2) = (-2)^2 - 2(-2) + 5 = 4 + 4 + 5 = 13

  • For x=5x = 5: f(5)=5225+5=2510+5=20f(5) = 5^2 - 2 \cdot 5 + 5 = 25 - 10 + 5 = 20

Step 4: Determine the Range

From the calculations above:

  • The minimum value of f(x)f(x) is 44, occurring at x=1x = 1.
  • The maximum value of f(x)f(x) on the interval [2,5][-2, 5] is 1313, occurring at x=2x = -2.

Thus, the range of f(x)f(x) for x[2,5]x \in [-2, 5] is: [4,13][4, 13]

Would you like further details on any specific part of the solution?

Here are some questions that expand on this solution:

  1. How can you find the range of a quadratic function for a different interval?
  2. What if the function opened downwards? How would the range change?
  3. How does completing the square help in finding the vertex of a quadratic function?
  4. What is the significance of the vertex form of a quadratic function?
  5. How would the process change if the function had a different coefficient for x2x^2?

Tip: Always check the endpoints of the interval when finding the range of a function restricted to a closed interval.

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Math Problem Analysis

Mathematical Concepts

Algebra
Quadratic Functions
Function Range

Formulas

Vertex form of a quadratic function
Range of a function on a closed interval

Theorems

Completing the square for quadratic functions

Suitable Grade Level

Grades 9-12