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Chapter 2: Problem 5

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

Short Answer

Expert verified
Yes, the relation defines a function.

Step by step solution

01

Understand the Definition of a Function

A relation defines a function if each input (x-value) maps to exactly one output (y-value). There should not be any x-value that corresponds to more than one y-value.
02

Identify the Input and Output Pairs

List out the given pairs: (-3,1), (4,1), (-2,7).
03

Check for Repeated Input Values

Check if any x-value is repeated with different y-values. The pairs are (-3, 1), (4, 1), (-2, 7). None of the x-values (-3, 4, -2) are repeated with different y-values.
04

Conclude if the Relation Defines a Function

Since no input (x-value) corresponds to more than one output (y-value), this 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
First, let鈥檚 explore the idea of a 'relation.' In the context of mathematics, a relation is a set of ordered pairs. An ordered pair consists of two elements: usually, the first element is an input (x-value) and the second element is an output (y-value).
In our example, the relation given is \[\{(-3,1),(4,1),(-2,7)\}\] .
This means we have three ordered pairs. But to understand if this relation is a function, we need to dive a bit deeper.
Input and Output
A good way to think about input and output is like a machine. You feed the machine (the function) an input (x-value), and it processes this input to give an output (y-value).
In our example, the inputs (x-values) are -3, 4, and -2. The outputs (y-values) are 1, 1, and 7.
For a relation to be considered a function, each input should correspond to exactly one output. This means that no input should be able to generate more than one output.
Let鈥檚 see if this holds for our relation.
X-Value and Y-Value
An ordered pairs鈥� input is the x-value while the output is the y-value. For the given example, we extract the x-values and the y-values from the ordered pairs:
- (-3,1): Here, -3 is the x-value, and 1 is the y-value.
- (4,1): Here, 4 is the x-value, and 1 is the y-value.
- (-2,7): Here, -2 is the x-value, and 7 is the y-value.

Now let鈥檚 check if any x-value repeats but pairs with a different y-value. As we see, -3, 4, and -2 are all unique x-values with each giving a unique y-value (1, 1, and 7 respectively). Thus, the condition holds that each x-value maps to only one y-value.
Definition of a Function
Finally, let鈥檚 conclude with the definition of a function. A function is a specific type of relation where every x-value (input) has a unique y-value (output).
This uniqueness means that no x-value should pair with more than one y-value. In our example, there are no repeated x-values with different y-values.
Each x-value maps to exactly one y-value. This satisfies the definition of a function.
Therefore, this relation indeed defines a function. So, the given set \[\{(-3,1),(4,1),(-2,7)\}\] is a function.

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Most popular questions from this chapter

Solve each problem. Relationship of Measurement Units The function defined by \(f(x)=12 x\) computes the number of inches in \(x\) feet, and the function defined by \(g(x)=5280 x\) computes the number of feet in \(x\) miles. What does \((f \circ g)(x)\) compute?

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 . $$f(x)=\sqrt{x}, \quad g(x)=x-1$$

Write an equation ( \(a\) ) in standard form and ( \(b\) ) in slope-intercept form for the line described. See Example \(6 .\) Find \(k\) so that the line through \((4,-1)\) and \((k, 2)\) is (a) parallel to \(3 y+2 x=6\) (b) perpendicular to \(2 y-5 x=1\)

For the pair of functions defined, find \((f+g)(x),(f-g)(x),(f g)(x),\) and \(\left(\frac{f}{g}\right)(x)\) Give the domain of each. $$f(x)=4 x^{2}+2 x, g(x)=x^{2}-3 x+2$$

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. $$(g \circ f)(6)$$
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