Math Problem Statement

Based on the query submitted by the user and the corresponding solution, generate the following information in JSON format.

Solution

To solve this problem, we need to identify the inverse points of the function f(x)f(x). For the inverse f1(x)f^{-1}(x), the xx- and yy-coordinates are swapped. So for each point (a,b)(a, b) on f(x)f(x), the corresponding point on f1(x)f^{-1}(x) is (b,a)(b, a).

Points from the graph:

  1. (2,4)(-2, -4)
  2. (1,2.25)(-1, -2.25)
  3. (0,2)(0, -2)
  4. (2,0)(2, 0)

Corresponding points for f1(x)f^{-1}(x):

  1. For (2,4)(-2, -4): inverse is (4,2)(-4, -2)
  2. For (1,2.25)(-1, -2.25): inverse is (2.25,1)(-2.25, -1)
  3. For (0,2)(0, -2): inverse is (2,0)(-2, 0)
  4. For (2,0)(2, 0): inverse is (0,2)(0, 2)

Table filled:

xxf1(x)f^{-1}(x)
4-42-2
2.25-2.251-1
2-200
0022

Let me know if you'd like further explanation or more examples!

Related Questions:

  1. What is the geometric relationship between a function and its inverse on the graph?
  2. How can you verify whether two points are inverses of each other mathematically?
  3. What happens to the domain and range when a function is inverted?
  4. Are all functions invertible? Why or why not?
  5. How does the line y=xy = x relate to the graph of a function and its inverse?

Tip:

For quick inversion, remember to interchange xx and yy, but ensure the function is one-to-one before assuming an inverse exists!

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Math Problem Analysis

Mathematical Concepts

Inverse Functions
Coordinate Geometry

Formulas

f^{-1}(x): Swap x and y coordinates

Theorems

Inverse of a function
Graphical relationship of functions and their inverses

Suitable Grade Level

Grades 9-12