Question

warm - ups 1/5 - 1/9
alayah shockley
tuesday, 1/6
determine which relations are functions or not functions.

1. a graph with a red circle centered at the origin on a coordinate grid.
2. a table with x values 0, 1, 2, 3 and corresponding y values 1, 2, 4, 8.
3. a mapping diagram with input 1, 2, 3 and output 4, 5, 6. input 1 maps to 4, input 2 maps to 5, input 3 maps to 6 (with some crossing arrows?)
4. a mapping diagram with input 1, 2, 3 and output 4, 5, 6. input 1 maps to 4, input 2 maps to 5, input 3 maps to 5 and 6 (two arrows from 3)

Explanation:

Step1: Recall the definition of a function

A relation is a function if each input (x - value) has exactly one output (y - value). We will check each relation one by one.

Step2: Analyze Relation 1 (Graph of a circle)

The graph is a circle centered at the origin. For a vertical line (e.g., \(x = 0\)), it intersects the circle at two points (\((0,r)\) and \((0, - r)\) where \(r\) is the radius). By the vertical line test, if a vertical line intersects the graph at more than one point, the relation is not a function. So, the circle is not a function.

Step3: Analyze Relation 2 (Table)

The table has \(x\) - values \(0,1,2,3\) and corresponding \(y\) - values \(1,2,4,8\). Each \(x\) - value has a unique \(y\) - value. So, this relation is a function.

Step4: Analyze Relation 3 (Input - Output Diagram)

The inputs are \(1,2,3\). Input \(1\) maps to \(4\), input \(2\) maps to \(5\), and input \(3\) maps to \(6\). Each input has exactly one output. So, this relation is a function.

Step5: Analyze Relation 4 (Input - Output Diagram)

The input \(3\) maps to both \(5\) and \(6\). Since one input has more than one output, this relation is not a function.

Answer:

1. Not a function (circle fails vertical line test)
2. Function (each \(x\) has unique \(y\))
3. Function (each input has one output)
4. Not a function (input 3 has two outputs)