Composition of Functions


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


c. g(g(−1))

Verified step by step guidance

1. First, identify the function g(x) that applies to the input value x = -1. Since -1 falls within the interval -2 ≤ x < 0, use the expression g(x) = -x.
2. Evaluate g(-1) using the expression g(x) = -x. Substitute -1 for x, which gives g(-1) = -(-1).
3. Simplify the expression -(-1) to find the value of g(-1).
4. Now, use the result from g(-1) as the new input for the function g(x) again. Determine which expression of g(x) applies to this new input value.
5. Evaluate g(g(-1)) using the appropriate expression for g(x) based on the new input value obtained from the previous step.

Key Concepts

Composition of Functions

Composition of functions involves combining two functions where the output of one function becomes the input of another. For example, if you have functions f(x) and g(x), the composition g(f(x)) means you first apply f to x, then apply g to the result. Understanding this concept is crucial for evaluating expressions like g(g(-1)).

Piecewise Functions

A piecewise function is defined by different expressions based on the input value. In this case, g(x) has two different rules depending on the value of x: one for -2 ≤ x < 0 and another for 0 ≤ x ≤ 2. Recognizing which part of the function to use based on the input is essential for correctly evaluating g(-1) and subsequently g(g(-1)).

Function Evaluation

Function evaluation is the process of finding the output of a function for a given input. This involves substituting the input value into the function's formula. For instance, to evaluate g(-1), you need to determine which piece of the piecewise function applies and then compute the result accordingly, which is a key step in solving the problem.