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A linear function is a type of mathematical function that can be represented by a straight line on a graph. It is a function that has a constant rate of change between its input and output values. This means that for every unit increase in the input, there is a constant increase in the output. Linear functions are used in a wide range of fields, from physics and economics to computer science and engineering.

To identify a table that represents a linear function, we need to look for certain characteristics. A linear function will have a constant rate of change between its input and output values. This means that the difference between any two consecutive output values will be the same. For example, if we have a linear function f(x) = 2x + 3, then the difference between f(1) and f(2) will be the same as the difference between f(2) and f(3). This constant rate of change is known as the slope of the function.

One way to identify a linear function is to look at the data in a table and see if the differences between the output values are constant. Let's take a look at two tables to illustrate this concept.

Table 1:

x | f(x) |

1 | 4 |

2 | 7 |

3 | 11 |

4 | 16 |

5 | 22 |

Table 2:

x | g(x) |

1 | 3 |

2 | 6 |

3 | 10 |

4 | 14 |

5 | 19 |

Table 1 represents a non-linear function because the differences between the output values are not constant. The difference between f(1) and f(2) is 3, the difference between f(2) and f(3) is 4, the difference between f(3) and f(4) is 5, and the difference between f(4) and f(5) is 6. This means that the slope of the function is changing, and therefore it is not a linear function.

On the other hand, Table 2 represents a linear function because the differences between the output values are constant. The difference between g(1) and g(2) is 3, the difference between g(2) and g(3) is 4, the difference between g(3) and g(4) is 4, and the difference between g(4) and g(5) is 5. This means that the slope of the function is constant, and therefore it is a linear function.

Another way to identify a linear function is to look at the graph of the function. A linear function will always form a straight line on a graph. The slope of the line will represent the constant rate of change between the input and output values.

Let's take the same function f(x) = 2x + 3 and graph it to see what a linear function looks like.

As we can see from the graph, the function forms a straight line with a positive slope. This means that for every unit increase in the input, the output increases by 2. This is the constant rate of change between the input and output values that is characteristic of a linear function.

In conclusion, a linear function is a type of mathematical function that has a constant rate of change between its input and output values. To identify a table that represents a linear function, we need to look for the constant differences between the output values. If the differences are constant.

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