Composition of Functions


Evaluate each expression using the functions
f(x) = 2 − x, g(x) = { −x, −2 ≤ x < 0
x − 1, 0 ≤ x ≤ 2


e. g(f(0))

Verified step by step guidance

1. First, identify the function f(x) = 2 - x. We need to evaluate f(0) first.
2. Substitute x = 0 into f(x): f(0) = 2 - 0.
3. Calculate f(0) to find the result, which will be used as the input for g(x).
4. Next, use the result from f(0) as the input for g(x). Determine which piece of the piecewise function g(x) to use based on the value of f(0).
5. Evaluate g(f(0)) by substituting the value of f(0) into the appropriate piece of g(x) and simplify the expression.

Key Concepts

Function Composition

Function composition involves combining two functions to create a new function. If you have two functions, f(x) and g(x), the composition g(f(x)) means you first apply f to x, and then apply g to the result of f. This process is essential for evaluating expressions where one function's output becomes the input for another.

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In the given problem, g(x) is defined differently for two intervals: one for values from -2 to 0 and another for values from 0 to 2. Understanding how to evaluate piecewise functions is crucial for correctly applying them in function composition.

Evaluating Functions

Evaluating a function involves substituting a specific value into the function's expression to find the output. For example, to evaluate f(0) in the function f(x) = 2 - x, you replace x with 0, resulting in f(0) = 2. This step is necessary before performing function composition, as it determines the input for the next function.