Question

unit d functions test review
look at the relation.
complete the statement about the relation.
the relation
a function because for every
there
exactly one
.
options: is, is not, x - value, y - value

Explanation:

Step1: Recall the definition of a function

A function is a relation where each \( x \)-value (input) is associated with exactly one \( y \)-value (output).

Step2: Analyze the given relation

Looking at the mapping:
- \( x = -2 \) maps to \( y = 1 \)
- \( x = 0 \) maps to \( y = 1 \) and \( y = 8 \)? Wait, no, wait the diagram: Wait, let's re - examine. Wait, the left oval is \( x \) ( - 2, 0, 2, 4) and right is \( y \) (1,8). Wait, \( x=-2\) → 1; \( x = 0\) → 1 and 8? Wait no, maybe I misread. Wait, the lines: \( - 2\) to 1, \( 0\) to 1 and 8? Wait no, maybe the diagram is: \( - 2\) → 1; \( 0\) → 1 and 8? Wait, no, the definition of a function is that each \( x \)-value has only one \( y \)-value. Wait, no, wait in the diagram, \( x = 0\) is connected to two \( y \)-values? Wait, no, maybe I made a mistake. Wait, no, let's check again. Wait, the problem's diagram: \( x=-2\) → 1; \( x = 0\) → 1 and 8? Wait, no, that can't be. Wait, no, maybe the lines are: \( - 2\) to 1; \( 0\) to 1 and 8? Wait, no, the definition of a function is that for every \( x \)-value, there is exactly one \( y \)-value. Wait, but in the diagram, if \( x = 0\) is connected to two \( y \)-values, then it's not a function. Wait, but maybe I misread. Wait, the options are "is", "is not", "x - value", "y - value". Wait, let's re - express: The relation [is/is not] a function because for every [x - value/y - value] there [is/is not] exactly one [x - value/y - value].

Wait, the correct definition: A function is a relation where each input (\( x \)-value) has exactly one output (\( y \)-value). So, if an \( x \)-value is mapped to more than one \( y \)-value, it's not a function. In the given mapping, if \( x = 0\) is mapped to two \( y \)-values (1 and 8), then the relation is not a function? Wait, no, maybe I misread the diagram. Wait, maybe the lines are: \( - 2\) → 1; \( 0\) → 1; \( 0\) → 8; \( 2\) → 8; \( 4\) → 8. So \( x = 0\) is mapped to two \( y \)-values (1 and 8). So, for the \( x \)-value of 0, there are two \( y \)-values. Therefore, the relation is not a function because for every \( x \)-value, there is not exactly one \( y \)-value? Wait, no, the question's options: the first blank: is / is not; second: x - value / y - value; third: is / is not; fourth: x - value / y - value.

Wait, let's do it step by step.

1. Recall the function definition: A function is a set of ordered pairs where each \( x \)-coordinate (input) is paired with exactly one \( y \)-coordinate (output). So, for a relation to be a function, every \( x \)-value must have only one \( y \)-value.

2. Analyze the mapping:
- \( x=-2\) is paired with \( y = 1\) (one \( y \)-value)
- \( x = 0\) is paired with \( y = 1\) and \( y = 8\) (two \( y \)-values)
- \( x = 2\) is paired with \( y = 8\) (one \( y \)-value)
- \( x = 4\) is paired with \( y = 8\) (one \( y \)-value)

Since the \( x \)-value \( 0\) is paired with two different \( y \)-values, the relation does not satisfy the definition of a function. So, the relation is not a function because for every \( x \)-value, there is not exactly one \( y \)-value? Wait, no, the options for the third blank are "is" or "is not"? Wait, the third blank is from the options "is" or "is not"? Wait, the options given are: "is", "is not", "x - value", "y - value".

So, let's structure the sentence:

The relation [is not] a function because for every [x - value] there [is not] exactly one [y - value]. Wait, no, the third blank: the options are "is" or "is not"? Wait, the options are:

First blank: "is" or "is not"

Second blank: "x - value" or "y - value"

Third blank: "is" or "is not"

Fourth blank: "x - value" or "y - value"

So, the correct sentence:

The relation is not a function because for every x - value there is not exactly one y - value.

Wait, but let's check again. The definition is: A function is a relation in which each input (x - value) has exactly one output (y - value). So, if an x - value has more than one y - value, the relation is not a function. So, in this case, since x = 0 has two y - values, the relation is not a function. So, the first blank: "is not", second blank: "x - value", third blank: "is not", fourth blank: "y - value". Wait, but the third blank's options are "is" or "is not"? Wait, the options are:

The buttons are: "is", "is not", "x - value", "y - value".

So, let's fill in:

The relation is not a function because for every x - value there is not exactly one y - value.

Wait, but the third blank: the word "there" is followed by "is" or "is not". So, "there is not exactly one y - value" for the x - value that has two y - values.

So, putting it all together:

The relation is not a function because for every x - value there is not exactly one y - value.

Answer:

The relation \(\boldsymbol{\text{is not}}\) a function because for every \(\boldsymbol{\text{x - value}}\) there \(\boldsymbol{\text{is not}}\) exactly one \(\boldsymbol{\text{y - value}}\).