Question

check for understanding: function vs not a function
determine if the following relation is a function. circle the correct choice. 1 point each

1. (1,4), (2,2), (3,8), (4,7)

function
not a function

1. table with x and y values

function
not a function

1. mapping diagram

function
not a function

1. graph

function
not a function

Explanation:

Problem 1: Relation \(\{(1,4.2), (2,2.1), (3,8.1), (4,7.1)\}\)

Step 1: Recall the definition of a function

A relation is a function if every input (x - value) has exactly one output (y - value).

Step 2: Check each x - value

• For \(x = 1\), the output is \(4.2\) (only one output).
• For \(x = 2\), the output is \(2.1\) (only one output).
• For \(x = 3\), the output is \(8.1\) (only one output).
• For \(x = 4\), the output is \(7.1\) (only one output).

Since each \(x\) - value has exactly one \(y\) - value, this relation is a function.

Problem 2: Vertical line test for the table (assuming it's a relation with \(x\) and \(y\) values, but the table seems to have \(x\) and repeated \(y\) - like values? Wait, the table has \(x\) as \(0,1,2,3,4,5\) and \(y\) as \(1,1,1,1,1,1\)? Wait, no, the original table: Let's re - examine. If the relation is such that for each \(x\) (input), how many \(y\) (outputs)? Wait, the table is presented as \(x\) with values \(0,1,2,3,4,5\) and \(y\) with the same value? Wait, no, maybe it's a relation where for each \(x\), there is one \(y\)? Wait, no, the problem says "Not a Function" is an option. Wait, maybe the table is a vertical line? Wait, no, if the relation is like \(x = 0\) maps to \(y = 1\), \(x = 1\) maps to \(y = 1\), \(x = 2\) maps to \(y = 1\), etc. Wait, no, the key is: A relation is not a function if an \(x\) - value has more than one \(y\) - value. If the table is a relation where, for example, if it's a vertical line (same \(x\) for multiple \(y\)s). Wait, the original problem's second part: the table - like structure. Wait, maybe it's a relation where one \(x\) has multiple \(y\)s. Wait, no, let's think again. The vertical line test: if we draw a vertical line on the graph of the relation, it should intersect the graph at most once. If the relation is a set of points where one \(x\) has multiple \(y\)s, it's not a function. Wait, the second problem: the table (probably a mapping where one \(x\) has multiple \(y\)s). Wait, maybe the table is a relation like \(x = 0\) is mapped to multiple \(y\)s? Wait, the problem's second part: the "Not a Function" option. Let's assume that in the second relation, an \(x\) - value is associated with more than one \(y\) - value. So, for example, if the relation is such that a single \(x\) has two or more \(y\)s, then it's not a function.

Problem 3: Mapping diagram (ovals)

Step 1: Recall the function definition for mappings

In a mapping from set \(X\) (domain) to set \(Y\) (codomain), a function requires that each element in \(X\) is mapped to exactly one element in \(Y\).

Step 2: Analyze the mapping

Looking at the mapping diagram, if there is an element in \(X\) that is mapped to more than one element in \(Y\), then it's not a function. From the diagram, if an \(x\) - element has multiple arrows to \(y\) - elements, then it's not a function. Let's assume that in the third diagram, an element in \(X\) is mapped to two elements in \(Y\), so it's not a function.

Problem 4: Graph (curve)

Step 1: Apply the vertical line test

The vertical line test states that a graph represents a function if and only if no vertical line intersects the graph at more than one point.

Step 2: Analyze the graph

Looking at the given graph, if we draw a vertical line, it will intersect the graph at more than one point (since it's a parabola - like curve opening to the left or right? Wait, if it's a curve that is not a function, like a parabola \(x=y^{2}\), then a vertical line \(x = a\) (\(a>0\)) will intersect the graph at two points \((a,\sqrt{a})\) and \((a,-\sqrt{a})\)). So, if the graph fails the vertical line test (a vertical line intersects it more than once), then it's not a function.

Answer:

s:

1. Function
2. Not a Function (assuming the table - like relation has an \(x\) with multiple \(y\)s)
3. Not a Function (assuming an \(x\) in the domain has multiple mappings)
4. Not a Function (fails vertical line test)