Invers Fungsi #Part 2 // Cara Menentukan Invers Fungsi // (Fungsi Linear Kuadrat Rasional Irasional)

Introduction

This tutorial will guide you through the process of finding the inverse of functions, focusing on linear, quadratic, rational, and irrational functions. Understanding how to determine the inverse of these functions is essential in mathematics, particularly in algebra and calculus, as it allows you to solve equations and understand function behavior.

Step 1: Understanding the Concept of Inverses

• An inverse function essentially reverses the operation of the original function.
• If you have a function f(x), its inverse is denoted as f⁻¹(x).
• The relationship between a function and its inverse can be expressed as:
  • f(f⁻¹(x)) = x
  • f⁻¹(f(x)) = x

Practical Tip

• To ensure a function has an inverse, it must be one-to-one (i.e., it passes the horizontal line test).

Step 2: Finding the Inverse of Linear Functions

• A linear function can be expressed in the form f(x) = mx + b.
• To find the inverse:
  1. Replace f(x) with y: y = mx + b
  2. Swap x and y: x = my + b
  3. Solve for y:
     • y = (x - b) / m
  4. Replace y with f⁻¹(x): f⁻¹(x) = (x - b) / m

Common Pitfall

• Ensure that the slope (m) is not zero; otherwise, the function does not have an inverse.

Step 3: Finding the Inverse of Quadratic Functions

• A quadratic function has the form f(x) = ax² + bx + c.
• To find the inverse, follow these steps:
  1. Replace f(x) with y: y = ax² + bx + c
  2. Swap x and y: x = ay² + by + c
  3. Solve for y:
     • Rearrange the equation into standard form and isolate y.
     • Use the quadratic formula if necessary:
       • y = [ -b ± √(b² - 4ac) ] / 2a
  4. Choose the appropriate root based on the domain of the original function.

Practical Tip

• Quadratic functions are not one-to-one over their entire domain; restrict the domain to ensure the function is one-to-one.

Step 4: Finding the Inverse of Rational Functions

• For a rational function f(x) = P(x)/Q(x), where P and Q are polynomials:
  1. Replace f(x) with y: y = P(x)/Q(x)
  2. Swap x and y: x = P(y)/Q(y)
  3. Solve for y:
     • Cross-multiply to eliminate the fraction.
     • Rearrange and isolate y.

Common Pitfall

• Check for values that would make Q(y) = 0, as these points will not be in the domain of the inverse.

Step 5: Finding the Inverse of Irrational Functions

• For functions involving square roots or other roots, such as f(x) = √(ax + b):
  1. Replace f(x) with y: y = √(ax + b)
  2. Swap x and y: x = √(ay + b)
  3. Square both sides to eliminate the root:
     • x² = ay + b
  4. Solve for y:
     • y = (x² - b) / a

Practical Tip

• Ensure that you consider the domain of the original function when determining the inverse.

Conclusion

In this tutorial, we covered how to find the inverse of various types of functions including linear, quadratic, rational, and irrational functions. Remember to consider the domain and ensure the function is one-to-one to successfully determine the inverse. As a next step, practice finding inverses for different functions to strengthen your understanding and application of these concepts.