Find \[$(g \circ F)(x)\$\] And \[$(f \circ G)(x)\$\] For The Given Functions \[$f\$\] And \[$g\$\].$\[ \begin{align*} f(x) &= \frac{4}{x+3}, \\ g(x) &= 3x-8 \end{align*} \\]$\[ (g \circ F)(x) = \square

Composition of Functions: Finding (g ∘ f)(x) and (f ∘ g)(x)

In mathematics, the composition of functions is a fundamental concept that allows us to combine two or more functions to create a new function. Given two functions f(x) and g(x), we can form two new functions: (g ∘ f)(x) and (f ∘ g)(x). In this article, we will explore how to find these composite functions for the given functions f(x) = 4/(x+3) and g(x) = 3x - 8.

Understanding Composition of Functions

The composition of functions is a way of combining two functions to create a new function. Given two functions f(x) and g(x), the composition of f and g is denoted by (g ∘ f)(x) or (f ∘ g)(x). The order of the functions matters, and the composition is not necessarily commutative.

Finding (g ∘ f)(x)

To find (g ∘ f)(x), we need to substitute f(x) into g(x). In other words, we replace x in g(x) with f(x).

Given f(x) = 4/(x+3) and g(x) = 3x - 8, we can substitute f(x) into g(x) as follows:

(g ∘ f)(x) = g(f(x)) = 3(f(x)) - 8

Now, we substitute f(x) = 4/(x+3) into the equation:

(g ∘ f)(x) = 3(4/(x+3)) - 8

To simplify the expression, we can multiply the numerator and denominator by (x+3):

(g ∘ f)(x) = 12/(x+3) - 8

We can rewrite the expression as a single fraction:

(g ∘ f)(x) = (12 - 8(x+3))/(x+3)

Simplifying the numerator, we get:

(g ∘ f)(x) = (12 - 8x - 24)/(x+3)

Combine like terms:

(g ∘ f)(x) = (-8x - 12)/(x+3)

Finding (f ∘ g)(x)

To find (f ∘ g)(x), we need to substitute g(x) into f(x). In other words, we replace x in f(x) with g(x).

Given f(x) = 4/(x+3) and g(x) = 3x - 8, we can substitute g(x) into f(x) as follows:

(f ∘ g)(x) = f(g(x)) = 4/((g(x))+3)

Now, we substitute g(x) = 3x - 8 into the equation:

(f ∘ g)(x) = 4/((3x - 8)+3)

Simplify the expression:

(f ∘ g)(x) = 4/(3x - 5)

In this article, we have found the composite functions (g ∘ f)(x) and (f ∘ g)(x) for the given functions f(x) = 4/(x+3) and g(x) = 3x - 8. We have shown that the composition of functions is a way of combining two functions to create a new function, and that the order of the functions matters.

(g ∘ f)(x) = (-8x - 12)/(x+3)

(f ∘ g)(x) = 4/(3x - 5)

The composition of functions is a fundamental concept in mathematics that has many applications in various fields, including physics, engineering, and economics. Understanding how to find composite functions is essential for solving problems in these fields.

• [1] "Composition of Functions" by Khan Academy
• [2] "Composition of Functions" by Math Open Reference
• [3] "Composition of Functions" by Wolfram MathWorld