Understanding the Composition of Functions: Finding (fg(x)) for (f(x) 3x^8) and (g(x) x - 2)

Function composition is a fundamental concept in mathematics, particularly in algebra. In this article, we will explore the process of finding the composition of two functions, specifically, how to determine (fg(x)) when given (f(x) 3x^8) and (g(x) x - 2).

What is (fg(x))?

Evaluation of (fg(x)) involves substituting the expression of one function into the other. Here, we want to find (fg(x)) using the functions (f(x) 3x^8) and (g(x) x - 2).

Steps to Solve for (fg(x))

Given Functions: We start with the two given functions: (f(x) 3x^8) (g(x) x - 2)

Step 1: Substitute (g(x)) into (f(x)).

(fg(x) f(g(x)) f(x - 2))

Step 2: Replace (x) in (f(x) 3x^8) with (x - 2).

(f(x - 2) 3(x - 2)^8)

Expanding and Simplifying the Expression

The process of expanding (3(x - 2)^8) to get (fg(x)) requires the use of the binomial theorem (if necessary) to expand the expression and then simplify it. However, in this case, we can directly replace (x) with (x - 2).

Therefore,

(fg(x) 3(x - 2)^8)

A subsequent step involves expanding (3(x - 2)^8), but for the purposes of this article, we will recognize that the simplified form of the expression is:

(fg(x) 3x - 6 times 8 3x^2)

However, the final simplified form, which is directly verifiable by direct substitution, is:

(fg(x) 3x^2)

Step-by-Step Calculation

(f(x) 3x^8) (g(x) x - 2) (fg(x) f(g(x)) f(x - 2)) (f(x - 2) 3(x - 2)^8) (3(x - 2)^8 3x - 6 times 8 3x^2) (fg(x) 3x^2)

Conclusion

In conclusion, finding the composition of functions (fg(x)) involves the careful substitution and simplification of expressions. The provided calculations clearly illustrate the process, and in this specific case, we arrive at: