Hello there, welcome to lesson seven. After completing this lesson, you should be able to identify a function and know the definition of a function. The idea of a function is at the heart of algebra. It is one of the most important concepts in mathematics today. So what is a function? Very simply, a function is a special kind of relation.
It's a rule that connects a set of inputs with a corresponding set of outputs, but with an important additional feature. If each input is related to one and only one output, then the relation is a function Here's a simple example of a function. The X values are 345 and six, and the corresponding y values are six 810 and 12. Notice that each x value is linked to one and only one y value. This means that the relation is a function. We can also see a pattern here.
Six equals two times three, eight equals two times 410 equals two times five, and 12 equals two times six. So the function is y equals two times x, or more simply y equals to x. Here's example one. Suppose y equals three x minus two, we first calculate the values of y for these values of x now Three, negative two, negative 1012, and three. Now when x is negative three, y is negative 11. When x is negative two, y is negative eight.
When x is negative one, y is negative five. When x equals zero, y equals negative two, when x equals one, y equals one, when x equals two, y equals four, and when x equals three, y equals seven. So here's the complete table, showing the pairs of x and y values. We can use this table to create an arrow diagram. list the x values, list the y values. Name the relation y equals three x minus two and use arrows to connect the related pairs of values.
Three and seven two and four, one and one, zero and negative two, negative one and negative five, negative two and negative eight, negative three and negative 11. Notice that each element of the first set is associated with one and only one element of the second set. So this means that y equals three x minus two is a function. Now the table can also be written as a set of ordered pairs like this, negative three and negative 11, negative two and negative eight, negative one and negative five, zero and negative two, one and one, two and four, and three and seven. Now an example two, we have not been given a rule or an equation, but we do have a table of values. X has the value Use 124 and five, and y has the corresponding values one, one, negative three and negative three.
This is what the arrow diagram looks like. Five is linked with negative three, four with negative three, two with one, and one with one. So the ordered pairs are one and one, two and one, four and negative three, and five and negative three. Notice that each x value is associated with one and only one y value. So this means that the relation shown by the table is a function. We've seen some relations that are functions.
Now let's look at some relations that are not functions. Example three shows the relation Why is this question A root of x. And here is the table of values for x and y. Notice the x values go from nine through zero and back to nine, and the y values from negative three through zero to positive three. For the input x equals naught, there is only one output, y equals naught. But when x equals one, there are two outputs, y equals negative one and y equals one.
And when x equals four, there are two outputs y equals negative two and y equals two. And for the input x equals nine, there are also two outputs, y equals negative three and y equals three. So this means that the relation y equals the square root of x is not a function. Look at this next Example, we're given a table of values. And we can see by inspection that when x equals two, there are two values of y, y equals one, and y equals three. So this means that the relation represented by this table of values is not a function.
This summarize what you've learned in this lesson, we can identify a function from a set of ordered pairs from a table of values. And from an arrow diagram. If each element of the first set is associated with one and only one element of the second set, then the relation is a function. more formal definition is a function is a rule that connects each element of the first set the x values with one and only one element of a second set the y values