Rumus Fungsi Invers - Matematika SMA Kelas XI Kurikulum Merdeka

Introduction

This tutorial will guide you through the concept of inverse functions as presented in the video "Rumus Fungsi Invers - Matematika SMA Kelas XI Kurikulum Merdeka." Understanding inverse functions is essential in mathematics, particularly in high school curriculum, as it lays the groundwork for more advanced topics. This guide will help you comprehend how to determine the inverse of a function using its formula.

Step 1: Understanding Functions and Their Inverses

• A function relates an input to a single output, often expressed as f(x).
• The inverse function, denoted as f⁻¹(y), reverses this relationship, meaning if f(a) = b, then f⁻¹(b) = a.
• Key terms to understand:
  • Domain: The set of all possible input values.
  • Codomain: The set of potential output values.
  • Range: The actual set of outputs from the function.

Step 2: Identifying if a Function Has an Inverse

• Not all functions have inverses. To determine if a function is invertible:
  • Check if it is one-to-one (bijective). This means each output corresponds to exactly one input.
  • Use the Horizontal Line Test: If any horizontal line crosses the graph of the function more than once, the function does not have an inverse.

Step 3: Finding the Inverse of a Function

• To find the inverse of a function given by its formula, follow these steps:
  1. Replace f(x) with y.
  2. Swap the roles of x and y. That is, replace y with x and x with y in the equation.
  3. Solve the new equation for y.
  4. Replace y with f⁻¹(x) to denote the inverse function.

Example

For the function f(x) = 2x + 3:

1. Replace f(x) with y: y = 2x + 3
2. Swap x and y: x = 2y + 3
3. Solve for y:
   • x - 3 = 2y
   • y = (x - 3) / 2
4. The inverse function is f⁻¹(x) = (x - 3) / 2.

Step 4: Verifying the Inverse Function

• To verify that you have found the correct inverse:
  1. Calculate f(f⁻¹(x)) and ensure it equals x.
  2. Calculate f⁻¹(f(x)) and ensure it equals x.
• If both conditions hold, your inverse function is correct.

Conclusion

In this tutorial, you learned about inverse functions, how to determine if a function has an inverse, and the process for finding and verifying the inverse function. Understanding these concepts not only enhances your mathematical skills but also prepares you for more complex topics. As a next step, practice finding inverses of different types of functions to solidify your understanding.