Komposisi Fungsi Part 3 - Fungsi invers dan Sifat-sifatnya [ Matematika Wajib Kelas X ]

Introduction

This tutorial covers the concepts of inverse functions, particularly focusing on linear and rational functions. Understanding inverse functions is essential in mathematics, especially in algebra and calculus, as it allows you to determine the function that reverses the effect of the original function.

Step 1: Understanding Inverse Functions

• Definition: An inverse function is a function that, when applied after the original function, returns the input value. If f(x) is a function, its inverse is denoted as f⁻¹(x).
• Key Property: The composition of a function and its inverse should yield the identity function:
  • f(f⁻¹(x)) = x
  • f⁻¹(f(x)) = x

Step 2: Finding the Inverse of Linear Functions

• General Formula: For a linear function of the form f(x) = ax + b, the steps to find its inverse are:
  1. Replace f(x) with y: y = ax + b
  2. Swap x and y: x = ay + b
  3. Solve for y:
     • y = (x - b) / a
  4. Rewrite the result as f⁻¹(x):
     • f⁻¹(x) = (x - b) / a
• Example: If f(x) = 2x + 3, then:
  • y = 2x + 3
  • x = 2y + 3
  • Solving gives f⁻¹(x) = (x - 3) / 2

Step 3: Finding the Inverse of Rational Functions

• General Approach: For a rational function of the form f(x) = (p(x) / q(x)):
  1. Set y = p(x) / q(x).
  2. Swap x and y: x = p(y) / q(y).
  3. Solve for y by cross-multiplying and isolating y.
• Example: For f(x) = (2x + 1) / (x - 1):
  • Let y = (2x + 1) / (x - 1)
  • Rearranging gives x(y - 1) = 2y + 1
  • Solve for y to find the inverse function.

Step 4: Properties of Inverse Functions

• Reflection: The graph of an inverse function is a reflection of the original function across the line y = x.
• Domain and Range: The domain of f becomes the range of f⁻¹ and vice versa.
• Existence of Inverses: Not all functions have inverses. A function must be one-to-one (bijective) to have an inverse.

Conclusion

In this tutorial, you learned how to define, find, and analyze inverse functions, focusing on linear and rational types. Understanding these concepts is crucial for deeper mathematical studies. For further practice, consider solving problems involving composition of functions and their inverses to reinforce your learning.