Matematika Teknik 4. Fungsi

Summary

This lecture covers concepts in Linear Algebra, specifically focusing on functions, composition of functions, and inverse functions. It revisits topics from high school mathematics, aiming to solidify students' understanding. The speaker explains the composition of functions, breaking down the substitution of variables step by step. The session also delves into the inverse of linear, quadratic, and rational functions, providing detailed explanations of each process. The lecture emphasizes mastering the fundamentals before using shortcuts and encourages students to follow structured mathematical methods.

Takeaways

• 📚 The next topic in Linear Algebra 2 is about functions, particularly composition functions.
• 🔁 Composition functions involve the combination of two functions sequentially, resulting in a new function.
• ✏️ In composition functions, when substituting a variable, every occurrence of the variable in the function is replaced accordingly.
• 💡 Understanding composition functions is crucial as many students still find it confusing despite its simplicity.
• 🔄 Inverse functions map a set from A to B, but not all functions have inverses, and some must be solved step by step.
• 📈 The process of finding the inverse of a linear function involves solving for 'x' in terms of 'y', reversing the function.
• ➗ Inverse functions for quadratic and rational functions require breaking down the equation, isolating variables, and using square roots or other techniques.
• 🧮 Special focus is placed on properly applying mathematical operations to both sides of an equation to maintain equality.
• ✍️ The process to find the inverse must be shown step by step in exams, rather than using the final inverse formula directly.
• ✅ Several examples demonstrate how to calculate the inverse for linear, quadratic, and rational functions step by step.

Q & A

What is the definition of function composition?

Function composition is the combination of two functions performed in sequence to produce a new function. It is represented as f(g(x)), where the output of function g becomes the input for function f.

How does the composition of two functions work in the example f(x) = 3x - 1 and g(x) = x + 4?

In this example, f(g(x)) is calculated by substituting g(x) into f(x). Since g(x) = x + 4, you substitute x + 4 into f(x), giving f(x + 4) = 3(x + 4) - 1, which simplifies to 3x + 12 - 1, resulting in 3x + 11.

What happens when the variable inside a function is replaced in function composition?

When a variable inside a function is replaced, all instances of that variable in the function are replaced with the new expression. For example, if f(x) = 3x - 1 and you replace x with (x + 4), the entire expression is adjusted accordingly.

What is the inverse function, and how is it defined?

An inverse function reverses the effect of the original function. If f(a) = b, then the inverse of f, denoted as f⁻¹, will map b back to a, so f⁻¹(b) = a. However, not all functions have inverses.

How is the inverse of a linear function like f(x) = ax + b found?

To find the inverse of a linear function f(x) = ax + b, you solve for x in terms of y by swapping f(x) with y and then isolating x. For example, y = ax + b becomes x = (y - b)/a, which gives the inverse function f⁻¹(x) = (x - b)/a.

What are the steps to find the inverse of a quadratic function?

To find the inverse of a quadratic function like f(x) = ax² + bx + c, you first set f(x) equal to y, then use completing the square or the quadratic formula to isolate x in terms of y. This process involves solving for x after rearranging the equation.

What is the significance of the variable's position in a function's inverse calculation?

In inverse calculation, the goal is to make the original independent variable (x) the subject of the equation. Any transformations applied to x must be undone to isolate it, ensuring that the equation reflects the reverse of the original function.

What are some examples of functions that do not have inverses?

Functions that are not one-to-one, meaning they map multiple inputs to the same output, do not have inverses. For example, many-to-one functions, such as quadratic functions without specific restrictions, do not have inverses across their entire domains.

How is the inverse of a rational function like f(x) = (ax + b)/(cx + d) derived?

To find the inverse of a rational function, you swap f(x) with y, then cross-multiply to eliminate the denominator. After rearranging the terms to group x on one side, you solve for x, and the resulting expression gives the inverse function.

What are common challenges students face when learning about function inverses?

Common challenges include understanding how to manipulate algebraic expressions, particularly when isolating variables in more complex functions like quadratics and rationals. Students may also struggle with recognizing which functions have inverses and how to apply inverse operations correctly.