Chapter 2, Problem 1CT

(a)

To determine

The range and domain of a relation $\{(2,5),(4,6),(6,7),(8,8)\}$ and whether the relation is a function or not.

Answer to Problem 1CT

Solution:

The domain of the relation is $\{2,4,6,8\}$ .

The range of the relation is $\{5,6,7,8\}$ .

The relation is a function.

Explanation of Solution

Given information:

The relation is $\{(2,5),(4,6),(6,7),(8,8)\}$ .

In the relation $R=\left\{\left(x,y\right)|x\in X,y\in Y\right\}$, the domain is the set of all inputs , $x$, and range is the set of all outputs , $y$ .

That is, in a set of order pairs, first coordinate of each order pair is an element of domain.

Therefore, the domain is $\{2,4,6,8\}$ .

Also, in a set of order pairs, second coordinate of each order pair is an element of range.

Therefore, the range is $\{5,6,7,8\}$ .

As output is different for each input, the relation is a function.

(b)

To determine

The range and domain of a relation $\{(1,3),(4,-2),(-3,5),(1,7)\}$ and whether the relation is a function or not.

Answer to Problem 1CT

Solution:

The domain of the relation is $\{1,4,-3\}$ .

The range of the relation is $\{3,-2,5,7\}$ .

The relation is not a function.

Explanation of Solution

Given information:

The relation is $\{(1,3),(4,-2),(-3,5),(1,7)\}$ .

In the relation $R=\left\{\left(x,y\right)|x\in X,y\in Y\right\}$, the domain is the set of all inputs, $x$, and range is the set of all outputs, $y$ .

In a set of order pairs, first coordinate of each order pair is an element of domain.

Therefore, the domain is $\{1,4,-3\}$ .

In a set of order pairs, second coordinate of each order pair is an element of range.

Therefore, the range is $\{3,-2,5,7\}$ .

For input $1$, there are two different outputs, $3\text{and}7$ .

Therefore, the relation $\{(1,3),(4,-2),(-3,5),(1,7)\}$ is not a function.

(c)

To determine

The range and domain of a relation and whether the relation is a function or not where the relation is

  

Answer to Problem 1CT

Solution:

The domain of the relation is $[-1,5]$ .

The range of the relation is $[-4,6]$ .

The relation is not a function.

Explanation of Solution

Given information:

The relation is:

  

Domain is set of $x$ values where the relation is defined.

Therefore, from graph, domain of relation is $[-1,5]$ .

The relation have minimum value at $y=-4$ and maximum value at $y=6$ .

Thus, the range of relation is $[-4,6]$ .

From graph, for $x\ge -1$, there are two different $y$ values for same input.

Thus, the relation is not a function.

(d)

To determine

The range and domain of a relation and whether the relation is a function or not where the relation is

  

Answer to Problem 1CT

Solution:

The domain of the relation is $[-5,5]$ .

The range of the relation is $\left[2,5\right)$ .

The relation is a function.

Explanation of Solution

Given information:

The relation is

  

Domain is set of $x$ values where the relation is defined.

Therefore, from graph, domain of relation is $[-5,5]$ .

The relation have minimum value at $y=2$ and maximum value below $y=5$ .

Thus, the range of relation is $\left[2,5\right)$ .

From graph, for each $x$ value, there is single $y$ value.

Thus, the relation is afunction.