Problem 7 - Introductory Algebra for College Students

Question

Determine whether each relation is a function. Give the domain and range for each relation. $$\\{(1,4),(1,5),(1,6)\\}$$

Step-by-Step Solution

Answer

The given relation is not a function. The domain is {1} and the range is {4, 5, 6}.

Step 1: Check if it’s a function

A relation is a function if each input (x-value) is related to exactly one output (y-value). In the given set, there are three ordered pairs: (1,4), (1,5), and (1,6). Each of these ordered pairs has the same x-value (1), but different y-values (4, 5, and 6). This means that one input (1) is related to more than one output, which violates the definition of a function. Therefore, the given relation is not a function.

Step 2: Determine the domain

The domain is the set of all possible x-values (inputs) in a relation. In the given set, the x-value is the same in all the ordered pairs: \(x=1\). Therefore, the domain of the given relation is {1}.

Step 3: Determine the range

The range is the set of all possible y-values (outputs) in a relation. In the given set, the y-values are different for all the ordered pairs: they are 4, 5, and 6. Therefore, the range of the given relation is {4, 5, 6}.