Chapter 10: Inverse Functions and Radicals

Overview

• Chapter focuses on inverse functions and radicals (square roots, cube roots, etc.).
• Includes review of exponent rules.
• Weighted assessment on inverses in the third nine weeks.
• Additional grades in the fourth nine weeks.
• Lecture notes do not match paper exactly; can refer to the key posted.

Power Functions

• Definition: Functions of the form (x^n).
• Function Definition:
  • Each (x) must be unique (cannot repeat).
  • Check using the vertical line test (disallow repeated (x) values).

Inverse Functions

• Finding Inverse:
  • Swap (x) and (y) in an equation.
  • Switch domain and range.
• Notation: (f^{-1}(x)).
• Example: Quadratic function (f(x) = x^2):
  • Sketch graph; find inverse by switching (x) and (y).
  • Resulting graph: a sideways parabola (not a function).
  • Fails the vertical line test.

Horizontal Line Test

• Determines if an inverse can be a function.
• Test: Original function fails horizontal line test, inverse is not a function.
• One-to-One (1:1) Functions:
  • Pass both vertical and horizontal line tests.
  • Examples: Linear, Cubic, Rational functions.

Domain Restrictions and Inverse Functions

• Quadratic Functions:
  • Restrict domain to make an invertible function.
  • Example: (f(x) = x^2), (x \geq 0).
  • Inverse: (y = \sqrt{x}) (positive square root).
• Graphical Reflection:
  • Original function and inverse are reflections over (y = x).

Inverse of Odd-Powered Functions

• Pass both tests without restriction.
• Example: (f(x) = x^3)
  • Inverse: (y = \sqrt[3]{x}) (cube root).
• Domain and Range: All real numbers.

Practical Application

• Finding inverses involves switching (x) and (y) and solving for (y).
• Be aware of domain and range restrictions based on function type.

Class Notes

• Reflecting over (y = x) used to verify inverse relationship graphically.
• Emphasized understanding the processes as they will recur in pre-calculus.
• Next class will involve transformations like shifting and reflections.

Note: Additional video will cover problem-solving steps for test preparation.