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What's a Function?

What's a function?

Summary

  1. You can think of a function as a souvenir penny machine -- you input a penny, and it outputs a new souvenir!
  2. A function can also be written in math terms
  3. In our example our input will be the variable 'x', which we can assign a value of 2
  4. The function, or operation, performed on the input 'x' is to add 3
  5. Then the function spits out a new output value, which is 5
  6. The input values for a function make up the domain, and the output values make up the range

Notes

    1. We'll make the input to the function 'x', a variable we'll assign a value of 2
    2. The function, or operation, is to add 3 to the input of 'x', or 2
    3. When we input 3 into the function, we get an output of 5
    4. So our function changed 3 into 5, just like the machine changed the normal penny into the flattened penny!
    1. Let's review the definition of a function, since we've looked at a real world example and a math example
    1. In other words, each value we put into our function will only have one possible output
    2. If we have more than one possible output value for a certain input, then we don't have a function!
    1. Some functions have many numbers in their domain, but others might only have a few
    2. If you get an answer that's undefined when you try and plug a value into a function, that value won't be in the domain
    1. Think of the range as the 'range' of output values that result from the input values in the domain
    1. A mapping diagram is one way to represent a function
    2. {7, 4, 3, -1} is the domain in this example
    3. {2, 0, 5} is the range
    4. In this example, the input 7 is assigned an output value of 0
    5. In this case we don't have an equation that defines our function -- we're just defining it using arrows
    6. The input 4 is assigned an output value of 2
    7. The input 3 is assigned an output value of 5
    8. The input -1 is assigned an output value of 5, as well
    1. Both input values of 3 and -1 produce 5 as an output
    2. So it's ok to have input values assigned to the SAME output value, as long as they are each assigned to only ONE output value
    1. Let's look at a mapping diagram example where an input value is assigned more than one ouput value
    2. {3, -2, 4} is the domain in this example
    3. {6, 1, 8} is the range in this example
    4. In this example, the input 3 is assigned an output value of 1
    5. The input -2 is assigned an output value of 6, as well as an output of value of 8
    6. The input 4 is assigned an output value of 1
    1. The input -2 is assigned an output value of 6, as well as an output of value of 8
    2. But in order to be a function, an input value can only have ONE output value
    3. So here we do NOT have a function!
    1. Usually a function rule is an equation of some kind
    2. Let's look back at the equation we used before
    3. Remember, we had an input value of x = 2
    4. The function, or operation, performed on the input 'x' is to add 3
    5. So then when we add 3 to 2, we get our output value of 5
    1. Each (x,y) in the set is an ordered pair
    2. The x-coordinates of each ordered pair represent the inputs of the function
    3. The y-coordinates of each ordered pair represent the outputs of the function
    4. Notice that this is just a different way of representing the function in our mapping diagram from before!
    1. Our set of ordered pairs is {(7,0), (4,2), (3,5), (-1,5)}
    2. The x-coordinates {7, 4, 3, -1} are the input values and make up the domain
    3. The y-coordinates {0, 2, 5, 5} are the output values and make up the range
    1. 'x' represents the input values for the function
    2. Sometimes you might see 'y' instead of 'f(x)' -- they mean the same thing, and both can represent the output values of the function
    3. {1, 2, 3, 4} make up the domain and {4, 5, 6, 7} make up the range
    4. You can also create tables if you have a rule
    5. In this case, the rule would be to add 3
    6. So this table also represents the function we had at the beginning!
    1. The green arrow on the graph represents a graphed function
    2. On a graph, the input values are the x-coordinates, which give the horizontal locations of points
    3. The output values are the y-coordinates, which give the vertical locations of points