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Chapter 2: Problem 55

Let \(f=\\{(2,6),(3,8),(4,5)\\}\) and \(g(x)=3 x+5 .\) Find the following. $$f(2)$$

Short Answer

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Step by step solution

01

- Understand the Definition of f

The function \(f\) is given by a set of ordered pairs: \(f = \{(2, 6), (3, 8), (4, 5)\}\). In this set, the first element of each pair is the input (or \(x\)-value) and the second element is the output (or \(y\)-value).
02

- Locate the Input

Find the pair in the set where the first element (input) corresponds to 2. Look for \((2, y)\) within the set \(\{(2, 6), (3, 8), (4, 5)\}\).
03

- Identify the Output

Once you have located the pair \((2, 6)\), identify the second element in this pair, which is the output. Here, the output corresponding to the input 2 is 6.

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Key Concepts

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

Ordered Pairs
Ordered pairs are a fundamental concept in mathematics. They are written in the form \((a, b)\), where \(a\) represents the input and \(b\) represents the output. In the context of functions, ordered pairs show the relationship between the input and the output. For example, in the function \(f = \{(2, 6), (3, 8), (4, 5)\}\), each ordered pair indicates a connection between an \(x\)-value (input) and a \(y\)-value (output). This relationship lets us evaluate functions by identifying the output that corresponds to a given input.
Input-Output Relationship
The input-output relationship in a function is crucial for understanding how functions operate. The input, often denoted as \(x\), is the value we plug into the function. The output, denoted as \(y\) or sometimes \(f(x)\), is the result of applying the function to the input. For instance, if we have the set of ordered pairs \(\{(2, 6), (3, 8), (4, 5)\}\), the input 2 yields the output 6, the input 3 yields the output 8, and the input 4 yields the output 5. To find an output for a given input, just locate the correct ordered pair and read the corresponding output.
Function Notation
Function notation is a shorthand way of expressing the relationship between inputs and outputs. Typically, functions are written using the notation \(f(x) = expression\), where \(f\) is the name of the function, and \(x\) is the input variable. For example, if \(f(x) = 3x + 5\), we can evaluate this function for any value of \(x\). To find \(f(2)\), we substitute 2 wherever \(x\) occurs in the expression, leading to the calculation \((3 \cdot 2) + 5 = 6 + 5 = 11\). For a set of ordered pairs, function notation helps identify which output corresponds to which input, like in the set \(\{(2, 6), (3, 8), (4, 5)\}\).

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