F(g(x))

Definition

The term f(g(x)) represents the composition of two functions, where the inner function g(x) is evaluated first and the result is then used as the input for the outer function f(x). This concept is central to understanding the topics of finding composite and inverse functions.

5 Must Know Facts For Your Next Test

1. The order of the functions in f(g(x)) is important, as the inner function g(x) is evaluated first, and its result is then used as the input for the outer function f(x).
2. To evaluate f(g(x)), you first need to find the value of g(x), and then use that value as the input for the function f(x).
3. The domain of f(g(x)) is the set of all x values for which both g(x) and f(g(x)) are defined.
4. Composite functions can be used to model complex real-world situations by combining simpler functions.
5. Finding the inverse of a composite function, f(g(x)), requires finding the inverse of both the inner function g(x) and the outer function f(x).

Review Questions

• Explain the step-by-step process for evaluating the expression f(g(x)).
  • To evaluate the expression f(g(x)), you first need to find the value of g(x) by substituting the given value of x into the inner function g(x). Once you have the value of g(x), you then use that value as the input for the outer function f(x). The final result is the value of f(g(x)). This process of evaluating the inner function first and then using its result as the input for the outer function is the key to understanding the composition of functions.
• Describe the relationship between the domain of the composite function f(g(x)) and the domains of the individual functions f(x) and g(x).
  • The domain of the composite function f(g(x)) is the set of all x values for which both g(x) and f(g(x)) are defined. This means that the domain of f(g(x)) is the intersection of the domain of g(x) and the domain of f(x), as the output of g(x) must be a valid input for f(x). Understanding the relationship between the domains of the individual functions and the composite function is crucial when working with composite functions.
• Explain the process of finding the inverse of a composite function f(g(x)).
  • To find the inverse of a composite function f(g(x)), you need to first find the inverse of the outer function f(x) and the inverse of the inner function g(x). The inverse of the composite function f(g(x)) is then given by the expression (f^-1)(g^-1)(x), where f^-1 is the inverse of f(x) and g^-1 is the inverse of g(x). This step-by-step process of finding the inverses of the individual functions and then composing them is the key to understanding how to find the inverse of a composite function.

Related terms

Composite Function: A composite function is a new function created by combining two or more existing functions, where the output of one function becomes the input of the next function.

Inverse Function: An inverse function is a function that 'undoes' the operation of another function, reversing the input and output values.

Domain and Range: The domain of a function is the set of all possible input values, while the range is the set of all possible output values. Understanding the domain and range of functions is crucial when working with composite and inverse functions.