Composite Functions

Definition and Formation of Composite Functions

Composite functions are formed by combining two functions such that the output of one function becomes the input of another. This process is denoted as (f \circ g)(x), which is read as "f composed with g." The composite function is defined by the equation:

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• Domain of Composite Function: The domain of f \circ g consists of all values x in the domain of g such that g(x) is in the domain of f.

• Notation: f \circ g means apply g first, then f to the result.

Additional info: The diagram above visually represents how the domain of g and the range of g interact with the domain of f to determine the domain of the composite function f \circ g.

Evaluating Composite Functions

To evaluate a composite function at a specific value, substitute the value into the inner function and then apply the outer function to the result.

• Step 1: Evaluate g(x).

• Step 2: Substitute g(x) into f to find f(g(x)).

Example: Suppose f(x) = x^2 and g(x) = x + 1. Find (f \circ g)(2):

• First, g(2) = 2 + 1 = 3.

• Then, f(3) = 3^2 = 9.

• So, (f \circ g)(2) = 9.

Finding the Domain of a Composite Function

Determining the domain of a composite function requires considering the domains of both functions and any restrictions imposed by their combination.

• Step 1: Identify the domain of g.

• Step 2: Find all x in the domain of g such that g(x) is in the domain of f.

Example: If f(x) = \frac{1}{x+4} and g(x) = \frac{1}{x-1}, the domain of g excludes x = 1 (division by zero). For f(g(x)), solve g(x) + 4 \neq 0 to find further restrictions.

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Solve for x to find additional values to exclude from the domain.

Examples of Composite Functions and Their Domains

• Example 1: If f(x) = x^2 and g(x) = x + 1, both have domain all real numbers, so f \circ g and g \circ f also have domain all real numbers.

• Example 2: If f(x) = \frac{1}{x-2} and g(x) = \frac{1}{x-1}, exclude x = 1 (from g) and x = 2 (from f), and solve for any additional restrictions from the composition.

Equality of Composite Functions

Two composite functions f \circ g and g \circ f are generally not equal. However, in some cases, they may be equal for all x in their common domain. To show equality, compute both compositions and compare their expressions.

Example: If f(x) = x + 2 and g(x) = x - 2, then:

• (f \circ g)(x) = f(g(x)) = g(x) + 2 = x - 2 + 2 = x

• (g \circ f)(x) = g(f(x)) = f(x) - 2 = x + 2 - 2 = x

• Thus, (f \circ g)(x) = (g \circ f)(x) = x for all x.

Decomposing Functions into Composites

Given a function H(x), it is often useful to express it as a composition of two simpler functions f and g such that H(x) = f(g(x)).

• Example: If H(x) = (x^3 + 1)^{21}, let g(x) = x^3 + 1 and f(x) = x^{21}. Then H(x) = f(g(x)).

• Example: If H(x) = \frac{1}{x^2 + 1}, let g(x) = x^2 + 1 and f(x) = \frac{1}{x}. Then H(x) = f(g(x)).

Summary Table: Steps for Composite Functions