David has a Master of Business Administration, a BS in Marketing, and a BA in History.
What Is Notation?
Notation is a system of signs or symbols that represent words, numbers, phrases, etc. Notation is often designed for a specific purpose and is very helpful in communicating words, numbers, and other things in an efficient manner.
A good example of notation is musical notation. This type of notation system contains many symbols and various types of graphics, and it allows a composer to inform others how he wants his or her music to sound and be performed. Musical notation can be intimidating when one is first exposed to it; however, once learned, it allows a musician or aspiring musician to write music and to learn the music of others.
Function notation is a system of symbols used in mathematics, but it is arguably much easier to learn than musical notation. Function notation is an efficient and effective way to display functions of all types in mathematics. Let's start with a linear function. Many students are familiar with seeing linear functions written as y = 3x + 7.
This is a linear function in slope-intercept form. It is written with the standard y-variable in terms of the x-variable, which is the input variable, or independent variable. This is generally how students first learn how to write functions in mathematics. It is a good way of exposing students to functions; however, y = 3x + 7 is not written in function notation. Let's call this function f. We can then rewrite it with function notation: f(x) = 3x + 7.
All we did is replace y with f(x), so y = f(x). We can read this symbol as the value of f at x or as f of x. This way of writing a function gives us more flexibility and a better way of communicating how a function needs to be evaluated.
Let's evaluate this function when x = 5. With function notation, the instructions are very simple: find f(5). This tells us to evaluate function f when x = 5. In other words, replace x with 5 and solve: f(5) = 3(5) + 7 = 15 + 7 = 22.
Let's try another example. Find f(7) for f(x) = x^2 - 10x: f(7) = 7^2 - 10(7) = 49 - 70 = -21.
Writing With Function Notation
Even though it is common to use the letter f when writing function notation, any letter can be used in either upper or lower case. We can also change the letter for the input variable, but it tends to stay in lower case. This flexibility provides valuable maneuverability in writing functions.
Let's say we wanted to write a function for the area of a square. We could call the function A for area and use the letter s for side length of the square. This would give us the function notation of A(s) = s^2. This notation tells us that the area of a square with side length of s is s^2.
Using this definition of function A, let's evaluate the following: A(6). A(6) = 6^2 = 36 square units.
We can use function notation for functions that have multiple input variables. The formula for the volume of a rectangular prism is V = lwh. The volume of a rectangular prism is a function of the prism's length (l), width (w), and height (h).
We can write this formula using function notation: V(l, w, h) = lwh. Let's evaluate V(3, 7, 2). The function notation tells us that the rectangular prism has a length of 3 units, a width of 7 units, and a height of 2 units: V(3, 7, 2) = (3)(7)(2) = 42 cube units.
As long as you are told how a function and its input variables are defined, function notation allows efficient communication of a problem.
Sometimes we are asked to evaluate a function by replacing the input variables with expressions that contain variables themselves. For instance, let's start with the function g(x) = x^2 + 5x - 8. Let's first evaluate g(3): g(3) = 3^2 + 5(3) - 8 = 9 + 15 - 8 = 16.
Next, we need to evaluate g($). This seems like an unusual request, but it is being used here to make a point. The function notation is just instructing us to replace each x in the function with a dollar sign: g($) = $^2 + 5$ - 8.
Now, let's evaluate g(x - 2). Students often struggle with this type of problem because there is still a variable for the input value; however, we just need to evaluate the original function by replacing each x with (x - 2): g(x - 2) = (x - 2)^2 + 5(x - 2) - 8.
Foil (x - 2)^2 and distribute the 5: x^2 - 4x + 4 + 5x - 10 - 8. Combine like terms: x^2 + x - 14. We still have a variable in our answer, but that is all right.
Function composition is the combining of two functions. Let f(x) = x + 10 and g(x) = x^2 - 1. The composition f of g is defined by the rule f(g(x)). This means that we need to replace each x in the function f with function g as follows: f(g(x)) = (x^2 - 1) + 10. We can simplify to get x^2 + 9. This is another way to show the composition of two functions.
Let's now find the composition of g of f: g(f(x)) = (x + 10)^2 - 1. We can simplify to get x^2 + 20x + 99 (note that the commutative property does not apply to function composition).
Using the same two functions, let's now find f(g(2)). This notation tells us to first evaluate g(2): g(2) = 2^2 -1 = 4 - 1 = 3.
We have shown that g(2) = 3. Therefore, we replace g(2) with 3 in the original problem and evaluate function f: f(g(2)) = f(3) = 3 + 10 = 13.
The input into function g was 2, and this gave us an output of 3. This output became the input into function f. Finally, function f gave us the output of 13.
Function notation is a refined and dignified method of naming and evaluating functions in the context of mathematics. At the same time, function notation is a simple concept that is easy to use. Recognizing and using function notation is a critical step in moving on to advanced mathematics, such as pre-calculus and calculus. Expect to see function notation in high school and college classes and even in the work place.
After this lesson, you'll have the ability to:
- Define function notation and identify its purpose
- Recall how to write formulas with function notation
- Explain how to evaluate functions
- Describe the process of function composition
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