Compare Properties of Functions Algebraically

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  • 0:01 Visualizing Functions
  • 0:48 Evaluating Functions
  • 2:48 Using Function Notation
  • 4:24 Lesson Summary
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Lesson Transcript
Instructor: John Sepanski

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.

Evaluating Functions

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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