John has taught 6th Grade Mathematics through Geometry and has a Master's degree in Education
In this lesson, we will explore algebraic and numeric properties and the evaluation of functions. We'll demonstrate function evaluations and the idea that every element of the domain has a unique element in the range.
Visualizing Algebraic Functions
Have you ever wondered how a gum ball machine works? When you place a quarter into a particular machine, it drops down 14 skittles or M&M's, etc. But, how does it know how to do that? Well, if you take apart a gum ball machine, you will find that there are a series of gears that turn based upon the coin actually being a quarter.
As the gears turn, they rotate a plastic tray that has a bin in it that will hold exactly 14 pieces of candy. When that bin rotates beneath an open slot in the candy jar, gravity does its job and 14 pieces drop into the slot. Then, the plastic tray continues to turn and drops the candy down the chute to the waiting hand of the buyer. Yum! Algebraic functions are like that.
When you evaluate a function for a given value, which we call domain, you obtain an output, which we call an element of the range. Domain is the set of input values of an algebraic equation. Range is the set of output values of an algebraic equation. In an algebraic function, there is exactly one input for every output. Let's look at an example:
y = 2x + 4
If you evaluate this expression for 1, 2, 3, 4, you get these outcomes:
y = 2(1) + 4 = 6
y = 2(2) + 4 = 8
y = 2(3) + 4 = 10
y = 2(4) + 4 = 12
Notice that for each input there is exactly one output. Let's try evaluating for 3 again. Let's make sure that the answer doesn't change. Wait a minute! Why would the answer ever change? Isn't an input domain = 3 always going to produce an output of range = 10? y = 2(3) + 4 = 6 + 4 = 10. 2(3) is always going to be 6, and 6 + 4 can never be anything but 10. For this reason, this algebraic expression is a function.
A function has exactly one distinct output value in the range for each input value of the domain. It's like true love! There is that special someone out there in the range that is simply perfect for the element of the range! When you have a function expression, y = 2x + 4, x is always the input (domain), and y is always the output (range).
Writing Functions Using Function Notation
Since this is a function, we can write the same equation using function notation: f(x) = 2x + 4. This means that the function of x, f(x), is equal to 2x + 4. 2 * x + 4.
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If we evaluate f(x) = 2x + 4 for f = 7, f(7) = 2x + 4, we simply input 7 into the domain value, x, and run the function machine. What happened inside of the machine? As the 7 is input into the machine, the gears start grinding. It's first multiplied by 2, and that product is 14. When we add 4 to that, we arrive at the output of 18. Therefore, f(7) = 18.
Let's put in a different value. Let's evaluate f(-2) for 2x + 4. When I put the 2 into the machine, I multiply 2(-2) and get a product of -4. When I add 4 to that, I get 0. Is there any other number that I can input into my function that will produce a range value of zero?
Conclusion: f(-2) is the only domain value that will produce the range value of zero. This is the true mathematical definition of a function. So, a function has exactly one, and only one, output for each input.
The domain is the set of input values of an algebraic equation. The range is the set of output values of an algebraic equation. If we have an algebraic expression, y = -2x + 1, we can evaluate that function for different values of x. For example, we can input values into the domain of 1, 2, 3, 4, and then turn on the function machine. If we evaluate the function, we input the values into the expression:
f(1) = -2(1) + 1
f(1) = -2 + 1
f(1) = -1
Remember, every value of the domain has exactly one output. Every value of the domain has exactly one value in the range. Thinking in the other direction, every value of the range does not have exactly one value of the domain. But that's another story! Right now we're finished with algebraic functions.
Once you are finished with this lesson, you should be able to:
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