Composition of Functions


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


f. f(g(1/2))

Verified step by step guidance

1. First, identify the function g(x) that corresponds to the input value x = 1/2. Since 0 ≤ 1/2 ≤ 2, use the piece of g(x) defined as g(x) = x - 1.
2. Substitute x = 1/2 into the function g(x) = x - 1 to find g(1/2).
3. Calculate g(1/2) by evaluating the expression 1/2 - 1.
4. Now, use the result from g(1/2) as the input for the function f(x). The function f(x) is defined as f(x) = 2 - x.
5. Substitute the value obtained from g(1/2) into f(x) to find f(g(1/2)). Evaluate f(2 - g(1/2)) to complete the composition of functions.

Key Concepts

Composition of Functions

The composition of functions involves combining two functions to create a new function. If you have two functions, f(x) and g(x), the composition f(g(x)) means you first apply g to x, and then apply f to the result of g. 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) has two different rules depending on the value of x: one for values between -2 and 0, and another for values between 0 and 2. Understanding how to evaluate piecewise functions is crucial for correctly applying them in compositions.

Function Evaluation

Function evaluation is the process of finding the output of a function for a specific input. For example, to evaluate g(1/2), you need to determine which piece of the piecewise function applies to 1/2 and then compute the output. This step is fundamental in solving the composition f(g(1/2)), as it requires accurate evaluation of each function involved.