FUNGSI, FUNGSI KOMPOSISI, FUNGSI INVERS – Materi Matematika Dasar UTBK SNBT dan SIMAK UI | Part.1

Introduction

This tutorial aims to provide a comprehensive understanding of functions, function composition, and inverse functions as covered in the video “FUNGSI, FUNGSI KOMPOSISI, FUNGSI INVERS – Materi Matematika Dasar UTBK SNBT dan SIMAK UI | Part.1.” These concepts are fundamental in mathematics and essential for preparations for the UTBK SBMPTN and SIMAK UI exams.

Step 1: Understanding Functions

• A function is a relation that assigns exactly one output (value) for each input (value).
• Functions can be denoted by letters like f, g, or h.
• Example: If f(x) = 2x + 3, then for input x = 2, the output is f(2) = 2(2) + 3 = 7.

Step 2: Identifying the Domain

• The domain is the set of all possible input values for a function.
• To determine the domain:
  • Look for restrictions such as division by zero or square roots of negative numbers.
  • Example: For f(x) = 1/(x-1), x cannot be 1 (division by zero).
• Commonly used intervals include:
  • All real numbers: (-∞, ∞)
  • Exclude specific points: (-∞, 1) ∪ (1, ∞)

Step 3: Exploring Examples of Functions

• Polynomial Functions: Functions that involve powers of x.
  • Example: f(x) = x^2 + 3x + 2
• Root Functions: Functions involving square roots.
  • Example: f(x) = √(x-1) has a domain of x ≥ 1.
• Rational Functions: Functions that are a ratio of two polynomials.
  • Example: f(x) = (x^2 + 1)/(x-2) with the restriction x ≠ 2.

Step 4: Function Composition

• Function composition involves combining two functions where the output of one function becomes the input of another.
• Notation: (f ∘ g)(x) = f(g(x)).
• Example:
  • Let f(x) = 2x and g(x) = x + 3.
  • Then, (f ∘ g)(x) = f(g(x)) = f(x + 3) = 2(x + 3) = 2x + 6.

Step 5: Understanding Inverse Functions

• An inverse function reverses the operation of the original function.
• Denoted as f^(-1)(x).
• To find the inverse:
  • Replace f(x) with y.
  • Swap x and y, then solve for y.
• Example:
  • For f(x) = 2x + 3, replace with y:
    • y = 2x + 3
    • x = 2y + 3
    • Solve for y: y = (x - 3)/2
  • Thus, f^(-1)(x) = (x - 3)/2.

Conclusion

Understanding functions, function composition, and inverse functions is crucial for success in mathematics, particularly for entrance exams like UTBK and SIMAK UI. Practice these concepts with various examples and exercises to strengthen your skills. For further study, consider exploring additional mathematical topics or review upcoming videos for more in-depth explanations and practice problems.