which of the following relations are functions? select all that apply. \n□ {(7,1),(9,2),(8,3),(1,1)}\n□…

which of the following relations are functions? select all that apply. \n□ {(7,1),(9,2),(8,3),(1,1)}\n□ {(8.1,1.8),(2.2,1.1),(7.3,3.7)}\n□ {\\left(1\\frac{3}{4},0\\right),\\left(\\frac{3}{14},-\\frac{1}{2}\\right),(7,-\\frac{1}{3}),\\left(\\frac{3}{14},1\\right)}\n□ {(0.4325,1.975),(7.111,89.02),(1093,-98765),(0.4325,0.0023)}\n□ {(9999,100000),(999,10000),(99,10000),(9,100)}\n□ {\\left(-\\frac{9}{10},\\frac{1}{2}\\right),\\left(2,\\frac{1}{2}\\right),\\left(\\frac{8}{11},\\frac{1}{2}\\right),\\left(\\frac{1317}{23},\\frac{1}{2}\\right)}

which of the following relations are functions? select all that apply. \n□ {(7,1),(9,2),(8,3),(1,1)}\n□ {(8.1,1.8),(2.2,1.1),(7.3,3.7)}\n□ {\\left(1\\frac{3}{4},0\\right),\\left(\\frac{3}{14},-\\frac{1}{2}\\right),(7,-\\frac{1}{3}),\\left(\\frac{3}{14},1\\right)}\n□ {(0.4325,1.975),(7.111,89.02),(1093,-98765),(0.4325,0.0023)}\n□ {(9999,100000),(999,10000),(99,10000),(9,100)}\n□ {\\left(-\\frac{9}{10},\\frac{1}{2}\\right),\\left(2,\\frac{1}{2}\\right),\\left(\\frac{8}{11},\\frac{1}{2}\\right),\\left(\\frac{1317}{23},\\frac{1}{2}\\right)}

Answer

To determine which relations are functions, we use the definition of a function: a relation is a function if each input (first element of the ordered pair) has exactly one output (second element of the ordered pair). In other words, no two ordered pairs in the relation can have the same first element with different second elements. Let's analyze each relation:

1. Relation: ({(7,1),(9,2),(8,3),(1,1)})

  • Check the first elements: (7, 9, 8, 1). All first elements are unique. So each input has exactly one output.
  • Conclusion: This is a function.

2. Relation: ({(8.1,1.8),(2.2,1.1),(7.3,3.7)})

  • Check the first elements: (8.1, 2.2, 7.3). All first elements are unique. So each input has exactly one output.
  • Conclusion: This is a function.

3. Relation: (\left{\left(1\frac{3}{4},0\right),\left(\frac{3}{14},-\frac{1}{2}\right),(7,-\frac{1}{3}),\left(\frac{3}{14},1\right)\right})

  • Check the first elements: (1\frac{3}{4}, \frac{3}{14}, 7, \frac{3}{14}). Here, the first element (\frac{3}{14}) appears twice with different second elements ((-\frac{1}{2}) and (1)).
  • Conclusion: This is not a function.

4. Relation: ({(0.4325,1.975),(7.111,89.02),(1093,-98765),(0.4325,0.0023)})

  • Check the first elements: (0.4325, 7.111, 1093, 0.4325). Here, the first element (0.4325) appears twice with different second elements ((1.975) and (0.0023)).
  • Conclusion: This is not a function.

5. Relation: ({(99999,1000000),(999,100000),(99,10000),(0.9,10)})

  • Check the first elements: (99999, 999, 99, 0.9). All first elements are unique. So each input has exactly one output.
  • Conclusion: This is a function.

6. Relation: (\left{\left(-\frac{9}{10},\frac{1}{2}\right),\left(2,\frac{1}{2}\right),\left(\frac{8}{11},\frac{1}{2}\right),\left(\frac{13}{23},\frac{1}{2}\right)\right})

  • Check the first elements: (-\frac{9}{10}, 2, \frac{8}{11}, \frac{13}{23}). All first elements are unique. Even though the second element is the same ((\frac{1}{2})) for all, the definition of a function allows multiple inputs to have the same output (it just prohibits multiple outputs for the same input).
  • Conclusion: This is a function.

Brief Explanations:

  • A relation is a function if each input (first element of ordered pair) has exactly one output (second element).
  • For ({(7,1),(9,2),(8,3),(1,1)}), all first elements are unique → function.
  • For ({(8.1,1.8),(2.2,1.1),(7.3,3.7)}), all first elements are unique → function.
  • For (\left{\left(1\frac{3}{4},0\right),\left(\frac{3}{14},-\frac{1}{2}\right),(7,-\frac{1}{3}),\left(\frac{3}{14},1\right)\right}), (\frac{3}{14}) has two outputs → not a function.
  • For ({(0.4325,1.975),(7.111,89.02),(1093,-98765),(0.4325,0.0023)}), (0.4325) has two outputs → not a function.
  • For ({(99999,1000000),(999,100000),(99,10000),(0.9,10)}), all first elements are unique → function.
  • For (\left{\left(-\frac{9}{10},\frac{1}{2}\right),\left(2,\frac{1}{2}\right),\left(\frac{8}{11},\frac{1}{2}\right),\left(\frac{13}{23},\frac{1}{2}\right)\right}), all first elements are unique (same output is allowed) → function.

Answer:

The relations that are functions are:

  • (\boldsymbol{{(7,1),(9,2),(8,3),(1,1)}})
  • (\boldsymbol{{(8.1,1.8),(2.2,1.1),(7.3,3.7)}})
  • (\boldsymbol{{(99999,1000000),(999,100000),(99,10000),(0.9,10)}})
  • (\boldsymbol{\left{\left(-\frac{9}{10},\frac{1}{2}\right),\left(2,\frac{1}{2}\right),\left(\frac{8}{11},\frac{1}{2}\right),\left(\frac{13}{23},\frac{1}{2}\right)\right}})