Composition of Functions


In Exercises 39 and 40, find


d. (g ○ g) (x).


ƒ(x) = 1/x , g(x) = 1/√ x + 2

Verified step by step guidance

1. First, understand the composition of functions. The notation (g ○ g)(x) means you need to apply the function g to itself, i.e., g(g(x)).
2. Start by substituting g(x) into itself. Since g(x) = 1/√(x + 2), replace x in g(x) with g(x) itself.
3. This substitution gives you g(g(x)) = 1/√(g(x) + 2). Now, substitute g(x) = 1/√(x + 2) into this expression.
4. You will have g(g(x)) = 1/√((1/√(x + 2)) + 2). Simplify the expression inside the square root.
5. Finally, simplify the entire expression to find the composition (g ○ g)(x). Remember to handle the square roots and fractions carefully during simplification.

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. The notation (f ○ g)(x) means applying function g to x first, and then applying function f to the result of g(x). This concept is essential for understanding how to evaluate expressions like (g ○ g)(x), where the function g is applied to itself.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. When composing functions, it is crucial to consider the domain of the inner function, as it can affect the overall composition. For example, in g(x) = 1/√(x + 2), the input x must be greater than or equal to -2 to avoid taking the square root of a negative number.

Evaluating Functions

Evaluating functions involves substituting a specific value into the function's formula to find the corresponding output. In the context of the question, evaluating (g ○ g)(x) requires first calculating g(x) and then substituting that result back into g. This step-by-step evaluation is fundamental for accurately determining the final output of the composed function.