Problem 99 - Precalculus Essentials

Question

Determine whether each relation is a function. Give the domain and range for each relation. a. \(\\{(1,6),(1,7),(1,8)\\}\) b. \(\\{(6,1),(7,1),(8,1)\\}\) (Section \(1.2,\) Example 2 )

Step-by-Step Solution

Answer

a. is not a function. The domain for a. is {1} and its range is {6,7,8}. However, b. is a function. Its domain is {6,7,8} and the range is {1}.

Step 1: Identify if a. is a function

The relation in part a is \(\{(1,6),(1,7),(1,8)\}\), which can be visually analyzed or checked. Here the same number in the domain, which is 1, corresponds to different numbers in the range (6,7,8), which contradicts the definition of a function. Therefore, relation a. is not a function

Step 2: Find the domain and range of a.

Though relation a. is not a function, its domain and range can be determined. The domain is the set of all first elements of the ordered pairs and it is {1}. The range is the set of all second elements of the ordered pairs, which is {6,7,8}

Step 3: Identify if b. is a function

The relation in part b is \(\{(6,1),(7,1),(8,1)\}\), which can be checked visually or analytically. Here different numbers in the domain (6,7,8), correspond to the same number in the range, which is 1. This is allowed in function, as one input corresponds to one output. Therefore, b is a function.

Step 4: Find the domain and range of b.

The domain for b. is the set of all first elements of the ordered pairs, {6,7,8}. The range is the set of all second elements of the ordered pairs, and it is {1}