For the following exercises, use each set of functions to...

For the following exercises, use each set of functions to find $f(g(h(x))) .$ Simplify your answers.
Given the functions $f(x)=\frac{1-x}{x}$ and $g(x)=\frac{1}{1+x^{2}},$ find the following:
a. $(g \circ f)(x)$
b. $(g \circ f)(2)$

Key Concepts

Algebraic Manipulation

Algebraic manipulation involves simplifying expressions through processes such as combining like terms, factoring, and reducing fractions. This skill is important when working with composite functions as it allows the complex expressions obtained from function composition to be simplified into a more manageable form.

Function Evaluation

Function evaluation is the process of determining the numerical value of a function given a specific input. When dealing with composite functions, evaluation typically requires first calculating the output of the inner function and then using this result as the input for the outer function, making accurate computation a vital part of the overall process.

Function Composition

Function composition is the process of applying one function to the result of another. It involves substituting one function into another, thereby creating a new function whose output depends on two or more functions. Understanding how to compose functions is essential in algebra, calculus, and many areas of mathematics because it allows the breakdown and combination of complex operations into simpler steps.

Substitution

Substitution is a method for evaluating functions where one expression is replaced by another equivalent expression. In the context of composite functions, substitution is used to insert the output of one function into another function's input, enabling the direct computation of the composite function.

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