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Chapter 9: Problem 36

For each function, find \((a) f(2)\) and \((b) f(-1)\) \(f=\\{(2,5),(3,9),(-1,11),(5,3)\\}\)

Short Answer

Expert verified
f(2) = 5, f(-1) = 11.

Step by step solution

01

- Understand the Function

The function is given as a set of ordered pairs: \{(2,5),(3,9),(-1,11),(5,3)\}. Each pair \(x, y\) represents the input \(x\) and the corresponding output \(y\).
02

- Find f(2)

To find \(f(2)\), look for the pair where the first element (input) is 2. In this case, the pair is \(2, 5\). Therefore, \(f(2) = 5\).
03

- Find f(-1)

To find \(f(-1)\), locate the pair where the first element (input) is -1. The pair is \(-1, 11\), so \(f(-1) = 11\).

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Key Concepts

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

ordered pairs
An ordered pair is a fundamental concept in mathematics. It consists of two elements or values, typically written in parentheses and separated by a comma, like this: \((x, y)\). The first value is called the input, while the second value is known as the output. Ordered pairs are essential in functions because they show the relationship between two variables.
In the given function \(\{(2,5),(3,9),(-1,11),(5,3)\}\), each pair represents a unique input and its corresponding output. For example, the ordered pair \((2, 5)\) indicates that when the input is 2, the output is 5.
input and output
The terms 'input' and 'output' are critical when understanding functions. The input is usually the value you provide to the function, and the function processes this value in some way to produce an output. In our example, the inputs and outputs are given directly as pairs.
For instance, if you have the input 2, you look for a pair where the first element is 2. The pair \((2, 5)\) shows that the output for this input is 5. Similarly, to find the output for an input of -1, locate the pair \(((-1, 11)\). Therefore, the output is 11.
function notation
Function notation \(f(x)\) is a way to name functions that clearly shows the relationship between the input and output. In \(f(x)\), \(x\) represents the input value, and \(f(x)\) denotes the output value.
For example, to find \(f(2)\) given the function \(\{(2,5),(3,9),(-1,11),(5,3)\}\), you look for the ordered pair where the first value is 2. Here, it's \((2, 5)\), so \(f(2) = 5\). To find \(f(-1)\), look for the pair with -1 as the first value. This pair is \((-1, 11)\), so \(f(-1) = 11\).
This functional notation is a concise and powerful way to describe the relationship between variables in mathematics.

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Most popular questions from this chapter

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ (g+h)\left(-\frac{1}{2}\right) $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{f}{g}\right)\left(\frac{3}{2}\right) $$

For each pair of functions, find \(\left(\frac{f}{g}\right)(x)\) and give any \(x\) -values that are not in the domain of the quotient function. $$ f(x)=2 x^{2}-x-3, \quad g(x)=x+1 $$

Let \(f(x)=x^{2}+4, g(x)=2 x+3,\) and \(h(x)=x-5 .\) Find each of the following. $$ (h \circ g)(4) $$

Let \(f(x)=x^{2}-9, g(x)=2 x,\) and \(h(x)=x-3 .\) Find each of the following. $$ \left(\frac{h}{g}\right)\left(-\frac{1}{2}\right) $$
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