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Chapter 6: Problem 12

Determine whether the following relations are functions. If the relation is not a function, explain why. $$\begin{array}{cc} \hline x & f(x) \\ \hline \frac{1}{2} & 1 \\ 1 & 2 \\ \frac{3}{2} & 1 \\ 2 & 2 \\ \frac{5}{2} & 3 \\ 3 & 3 \\ \hline \end{array}$$

Short Answer

Expert verified
Yes, the relation is a function; each input has a unique output.

Step by step solution

01

Understand the definition of a function

A relation is a function if every input value (from the domain) is associated with exactly one output value (in the range). This means that each 'x' value can only map to one 'f(x)' value in the table.
02

Analyze the given relation

Examine each pair in the given table to check if any 'x' value appears more than once with different 'f(x)' values. Our table has the pairs: \( (\frac{1}{2}, 1), (1, 2), (\frac{3}{2}, 1), (2, 2), (\frac{5}{2}, 3), (3, 3) \).
03

Check each input value for unique outputs

Go through each of the 'x' values: \( \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, 3 \). Ensure that none of the x-values are repeated with different f(x) values. Here, each x-value is associated with exactly one f(x) value.
04

Conclusion

Since every 'x' maps to a unique 'f(x)' and no x-value is repeated with a different output, the relation qualifies as 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 describes a set of ordered pairs, often representing a link between two sets of values. Consider a case where you have input values and output values grouped into pairs, similar to the example from the exercise:
  • Pairs like \(\left( \frac{1}{2}, 1 \right)\) and \(\left( 1, 2 \right)\) are part of a relation.
  • These pairs show how each input value is related to a corresponding output value.

A relation itself does not need to follow any specific rules. However, when working with functions, we impose extra conditions that every input value, or element from the domain, maps to exactly one output value in the range.
Input-Output Mapping
The concept of input-output mapping is central to understanding what makes a relation a function. For a relation to qualify as a function:
  • Each input (x-value) must map to only one output (f(x)-value).

Think of input-output mapping like a vending machine:
  • You select an item by pressing a button (input).
  • The machine should consistently provide the same designated snack or drink (output).
If you press the same button, you expect the same result each time. Similarly, in a function, each input corresponds to precisely one output. Our example exercise showcases a list of input-output pairs where each input value maps clearly to a unique output. This consistent mapping confirms the relation as a function.
Domain and Range
The domain and range of a relation are foundational to how we understand its potential as a function.
  • The domain refers to the complete set of possible inputs (x-values).
  • For the given relation, the domain is \(\left\{ \frac{1}{2}, 1, \frac{3}{2}, 2, \frac{5}{2}, 3 \right\}\).
  • The range is the set of all possible outputs (f(x)-values).
  • In our example, the range is \(\left\{ 1, 2, 3\right\}\).

The connection between domain and range can be visualized by thinking of them as the ingredients (domain) and the resulting dishes (range) in a recipe book. Just like each set of ingredients leads to one particular recipe, each input in the domain of a function maps uniquely to one output in the range. This one-to-one relationship is what guarantees that a relation is indeed a function.

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Most popular questions from this chapter

Consider the following scenario: Sales of the new RoboYak talking Internet assistant have been increasing since its debut a year and a half ago. The rate of increase in sales can be modeled by the rate function $$f(x)=\left\\{\begin{array}{cl} 6500, & 0 \leq x \leq 18 \\ 0, & \text { otherwise } \end{array}\right.$$ where \(x\) represents the number of months since the debut of the device and \(f(x)\) represents the rate of change in sales measured in units per month. How many units were sold from 12 to 18 months?

Find the area under \(f(x)\) on the indicated interval. Round the area to two decimal places as necessary. $$f(x)=\left\\{\begin{array}{cl} 130.6-9.6 x, & 5 \leq x \leq 12 \\ 0, & \text { otherwise } \end{array} \text { on the interval } 6 \leq x \leq 10\right.$$

Determine the area under each constant function on the indicated interval. Then graph the result. $$P(x)=\left\\{\begin{array}{ll} \frac{1}{8}, & 0 \leq x \leq 8 \\ 0, & \text { otherwise } \end{array} \text { on the interval } 2 \leq x \leq 5\right.$$

Consider the following scenario: The Pick-Chick restaurant charges \(\$ 2.50\) per chicken piece sold on the first 5 pieces and \(\$ 2.00\) per piece thereafter up to 10 pieces. The cost of the chicken is expressed by the piece wise defined function \(f(x)=\left\\{\begin{array}{cl}2.50 x, & x=1,2,3,4,5 \\\ 2.5+2 x, & x=6,7,8,9,10\end{array}\right.\) where \(x\) represents the number of pieces of chicken sold and \(f(x)\) represents cost in dollars. Evaluate \(f(4)\) and interpret the result.

Solve the following applications involving the area of a rectangle. A television screen has a width of 36.6 inches and a height of 20.6 inches. What is the area of the screen?
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