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Chapter 2: Problem 1

Decide whether each relation defines a function. $$\\{(5,1),(3,2),(4,9),(7,8)\\}$$

Short Answer

Expert verified
Yes, the relation defines a function.

Step by step solution

01

- Understand the definition of a function

A relation is a function if every input (first element of each pair) maps to exactly one output (second element of each pair). This means that no input value can correspond to more than one output value.
02

- Identify the inputs

List the first elements (inputs) from each ordered pair: 5, 3, 4, 7.
03

- Check for repeated inputs

Examine the list of inputs. If any input appears more than once, the relation is not a function. The inputs here are 5, 3, 4, and 7, with no repeats.
04

- Conclude

Since each input maps to a unique output and no input is repeated, the given relation defines a function.

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Key Concepts

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

Relation
In mathematics, a relation is a set of ordered pairs. Think of it like a collection of these pairs, where each pair consists of two elements: the first element (input) and the second element (output). Here鈥檚 a simple way to understand it:
Let's take the set given in the exercise: \(\{(5, 1), (3, 2), (4, 9), (7, 8)\}\).
Each pair involves a number mapping from one thing to another:
  • (5, 1): 5 maps to 1
  • (3, 2): 3 maps to 2
  • (4, 9): 4 maps to 9
  • (7, 8): 7 maps to 8

This collection is our relation, showing how different elements are connected. But, as we鈥檒l see next, not all relations qualify as functions.
Function Definition
A function is a special kind of relation. For a relation to be a function, every input must map to exactly one output. This means:
  • Each first element (input) in a pair is connected to only one second element (output).
  • No input value can have more than one output value associated with it.

Using our set from the exercise again: \(\{(5, 1), (3, 2), (4, 9), (7, 8)\}\), let's see if it qualifies as a function. We list the inputs: 5, 3, 4, and 7. There are no repeated inputs, and each input has exactly one output. This means our relation is indeed a function.
In summary, every element from the input set has a unique partner in the output set, fulfilling the criteria for a function.
Input-Output Mapping
The process of connecting each input to its corresponding output is known as input-output mapping. Imagine you have a machine where you insert an input and it gives you an output. If the machine gives you more than one output for a single input, it's broken. Similarly, in our case:
For the set \(\{5 \to 1, 3 \to 2, 4 \to 9, 7 \to 8\}\):
  • Input 5 gives output 1
  • Input 3 gives output 2
  • Input 4 gives output 9
  • Input 7 gives output 8

Each input has one clear and unique output, which makes it easy for us to confirm that this relation defines a function.
So, to solidify the concept: a function perfectly maps each input to one and only one output in a consistent way. This is the core idea behind determining whether a relation is a function or not.

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Most popular questions from this chapter

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{|c|c|c|c|}\hline x & 3 & 4 & 6 \\\\\hline f(x) & 1 & 3 & 9 \\\\\hline\end{array}$$ $$\begin{array}{|c|c|c|c|c|}\hline x & 2 & 7 & 1 & 9 \\\\\hline g(x) & 3 & 6 & 9 & 12 \\\\\hline\end{array}$$ Find each of the following. $$(g \circ f)(6)$$

Solve each problem. A software author invests his royalties in two accounts for 1 yr. (a) The first account pays \(4 \%\) simple interest. If he invests \(x\) dollars in this account, write an expression for \(y_{1}\) in terms of \(x,\) where \(y_{1}\) represents the amount of interest earned. (b) He invests in a second account $$ 500\( more than he invested in the first account. This second account pays \)2.5 \%\( simple interest. Write an expression for \)y_{2}\( where \)y_{2}\( represents the amount of interest earned. (c) What does \)y_{1}+y_{2}\( represent? (d) How much interest will he receive if $$ 250\) is invested in the first account?

Use a graphing calculator to solve each linear equation. $$7 x-2 x+4-5=3 x+1$$

Given functions \(f\) and \(g,\) find ( \(a\) ) \((f \circ g)(x)\) and its domain, and ( \(b\) ) \((g \circ f)(x)\) and its domain. See Examples 6 and 7 . $$f(x)=\sqrt{x}, \quad g(x)=\frac{1}{x+5}$$

For each of the functions in Exercises \(33-46,\) find ( \(a\) ) \(f(x+h),\) (b) \(f(x+h)-f(x)\) and \((c) \frac{f(x+h)-f(x)}{h} .\) See Example 4. $$f(x)=-x^{2}$$
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