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Chapter 2: Problem 33

For each function, find (a) \(f(2)\) and (b) \(f(-1)\) $$ f=\\{(-2,2),(-1,-1),(2,-1)\\} $$

Short Answer

Expert verified
(a) f(2) = -1, (b) f(-1) = -1

Step by step solution

01

Understand the function as a set of pairs

The function is given as a set of ordered pairs \{(-2,2),(-1,-1),(2,-1)\}. Each pair represents an input-output relationship, \((x, f(x))\).
02

Find f(2)

Look for the pair where the first element (input) is 2. That pair is \((2, -1)\). Therefore, \(f(2) = -1\).
03

Find f(-1)

Look for the pair where the first element (input) is -1. That pair is \((-1, -1)\). Therefore, \(f(-1) = -1\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ordered Pairs
Functions can be seen as a collection of ordered pairs. Each pair consists of two elements: an input and an output. For instance, the function \{(-2, 2), (-1, -1), (2, -1)\} includes three ordered pairs. Here, each pair follows the format \((x, f(x))\), where 'x' represents the input and 'f(x)' represents the corresponding output.
Understanding ordered pairs is crucial because it allows us to see how different inputs are mapped to their outputs within the function.
Input-Output Relationship
In mathematics, functions describe the relationship between inputs and outputs. This relationship is often written as \((x, f(x))\), where 'x' is the input value and 'f(x)' is the output value derived from the function rule.
For example, in the set \{(-2, 2), (-1, -1), (2, -1)\}, the pair \((2, -1)\) shows that when 2 is input into the function, the output is -1. Similarly, the pair \((-1, -1)\) indicates that an input of -1 also results in an output of -1.
This input-output relationship helps us predict the output when given a specific input.
Function Notation
Function notation is a way to describe functions in a precise manner. We use symbols like 'f(x)' to denote the function value at 'x'. For instance, if we want to find the value of the function at x=2, we write 'f(2)'. The process involves finding the ordered pair with the first element equal to 2.
In the given function \{(-2, 2), (-1, -1), (2, -1)\}, locating the pair \((2, -1)\) means that f(2) = -1. Similarly, for 'f(-1)', we look for the pair \((-1, -1)\), indicating that f(-1) = -1.
Function notation provides an efficient way to communicate and evaluate functions without having to describe the entire process each time.

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