Math Problem Statement

Solution

To solve the given problem, we will proceed as follows:

(i) Find f(1)f(-1) and f(3)f(3)

The function given is f(x)=x210x+16f(x) = x^2 - 10x + 16.

  1. Finding f(1)f(-1): f(1)=(1)210(1)+16=1+10+16=27f(-1) = (-1)^2 - 10(-1) + 16 = 1 + 10 + 16 = 27

  2. Finding f(3)f(3): f(3)=(3)210(3)+16=930+16=5f(3) = (3)^2 - 10(3) + 16 = 9 - 30 + 16 = -5

(ii) State the range of f(x)f(x)

To find the range of f(x)=x210x+16f(x) = x^2 - 10x + 16 for 2<x<82 < x < 8, we will:

  1. Complete the square to find the vertex: f(x)=x210x+16=(x5)225+16=(x5)29f(x) = x^2 - 10x + 16 = (x - 5)^2 - 25 + 16 = (x - 5)^2 - 9 The vertex of the parabola (x5)29(x - 5)^2 - 9 is at x=5x = 5 and f(5)=9f(5) = -9.

  2. Evaluate f(x)f(x) at the endpoints of the interval 2<x<82 < x < 8:

    • At x=2x = 2: f(2)=2210(2)+16=420+16=0f(2) = 2^2 - 10(2) + 16 = 4 - 20 + 16 = 0
    • At x=8x = 8: f(8)=8210(8)+16=6480+16=0f(8) = 8^2 - 10(8) + 16 = 64 - 80 + 16 = 0

Since the parabola opens upwards, the minimum value within 2<x<82 < x < 8 occurs at the vertex x=5x = 5, and we have evaluated the function at the endpoints of the interval to find the maximum values. Therefore, the range of f(x)f(x) for 2<x<82 < x < 8 is: [9,0][-9, 0]

Summary

(i) f(1)=27f(-1) = 27 and f(3)=5f(3) = -5

(ii) The range of f(x)f(x) for 2<x<82 < x < 8 is [9,0][-9, 0].

Would you like more details on any of these steps, or do you have any questions?

Related Questions

  1. What is the vertex form of a quadratic function?
  2. How do you complete the square for a quadratic equation?
  3. How can you determine the range of a quadratic function on a given interval?
  4. What are the properties of parabolas that open upwards?
  5. How do you find the maximum and minimum values of a function on a closed interval?
  6. What is the significance of the vertex in a parabola?
  7. How do transformations affect the graph of a quadratic function?
  8. Can you explain how to solve quadratic equations by factoring?

Tip

When finding the range of a quadratic function within a specific interval, always evaluate the function at the endpoints and the vertex if it lies within the interval.

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Math Problem Analysis

Mathematical Concepts

Quadratic Functions
Vertex Form
Completing the Square
Range of Functions

Formulas

Quadratic function formula f(x) = ax^2 + bx + c
Vertex form of a quadratic function

Theorems

Properties of parabolas

Suitable Grade Level

Grades 10-12