Computing The Composition Of Two Functions Section 16
Computing The Composition Of Two Functions Section 16 And Computing
Computing the composition of two functions (Section 1.6) and computing the inverse of a function (Section 1.8), are two of the most important topics we are covering in this module. These algebraic procedures are important in theory and in practice and are relevant in the study of calculus. This discussion activity will help you to become proficient in these skills. Create and post two functions, f(x) and g(x), of the types we have considered so far in this course. Make the functions challenging, but not beyond what is covered in the textbook; avoid very simple examples.
Paper For Above instruction
In this paper, I will demonstrate the computation of the composition of two functions and explore the inverse of these functions, illustrating their significance in mathematical analysis and calculus. To meet the challenge and adhere to the course principles, I will select functions that are sufficiently complex—more than simple linear or quadratic forms—yet within the scope of the course content.
The functions chosen for this exercise are:
- f(x) = 2x^3 - 5x + 4
- g(x) = √(x + 3) - 2
These functions were selected because they involve polynomial and radical expressions, adding a level of complexity suitable for the course level and thus suitable for demonstrating composition and inversion procedures.
Composition of the Functions
The composition of two functions, denoted by (f ⚬ g)(x), involves substituting g(x) into f(x). First, I will compute g(x), then substitute it into f(x).
Calculating g(x):
g(x) = √(x + 3) - 2.
Now, the composite function (f ⚬ g)(x) is calculated by replacing every occurrence of x in f(x) with g(x):
f(g(x)) = 2 [g(x)]^3 - 5 g(x) + 4.
This yields:
f(g(x)) = 2 [√(x + 3) - 2]^3 - 5 [√(x + 3) - 2] + 4.
This expression, though complex, demonstrates the process of composition, involving radical and polynomial expressions, and can be expanded further if needed for detailed analysis.
Inversion of the Functions
Next, I explore the inverse functions, which reverse the role of inputs and outputs. Starting with g(x), I aim to find g^(-1)(x).
Given g(x) = √(x + 3) - 2, I replace g(x) with y for convenience:
y = √(x + 3) - 2.
To find g^(-1)(x), I solve for x in terms of y:
- y + 2 = √(x + 3)
- Square both sides: (y + 2)^2 = x + 3
- Thus, x = (y + 2)^2 - 3
Replacing y with x in the last expression gives the inverse function:
g^(-1)(x) = (x + 2)^2 - 3.
This inverse function is valid within the domain that maintains the original function's range and ensures the inverse's correctness, emphasizing the importance of domain restriction.
Conclusion
This exercise underscores the importance of understanding function composition and inversion, key concepts in algebra and calculus. Through managing these operations with functions like polynomial and radical expressions, students develop analytical skills critical for higher mathematics. The choice of functions provides a challenge appropriate to the course level and emphasizes the need for careful algebraic manipulation and recognition of domain considerations in inverse functions.
References
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