Function Notation & Evaluating at Numbers
Purplemath
You've been playing with "y=" sorts of equations for some time now. And you've seen that the "nice" equations (straight lines, say, rather than ellipses) are the ones that you can solve for "y=" and then plug into your graphing calculator.
These "y=" equations are functions.
These "y=" equations are functions. But the question you are facing at the moment is "Why do I need this function notation, especially when I've got perfectly nice 'y=', and how does this notation work?"
What is function notation?
Function notation gives a name to a function, and makes clear which variable is independent (that is, which is the variable for which you'll be picking input values). The name for the formula, such as A (for, say, the area of a square), is followed by a parenthetical, such as (s) (for the length of a side of the square), and is set equal to the formula. In this example, instead of saying y = s2, the function notation for the exact same thing would be A(s) = s2.
How do you write function notation?
Function notation is written as follows:
[function name]([input variable]) = [formula]
For instance, the formula for the circumference C of a circle, in terms of the value of the circle's radius r, can be written as C = 2πr or, in function notation, it can be written as C(r) = 2πr.
Why do we write in function notation?
Instead of naming every function "y", function notation allows for a variety of names, such as f(x) and g(x), much as your calculator names the various formulas you've entered as Y1, Y2, and so forth. But function notation also allows you to point out the input variable — being the x inside the parentheticals of f (x) and g(x).
Think back to when you were in elementary school. Your teacher gave you worksheets containing statements like "[ ] + 2 = 4" and told you to fill in the box. Once you got older, your teacher started giving you worksheets containing statements like "x + 2 = 4" and told you to "solve for x".
Why did your teachers switch from boxes to variables? Well, think about it: How many shapes would you have to use for formulas like the one for the area A of a trapezoid with upper base a, lower base b, and height h? The formula is as follows:
If you try to express the above, or something more complicated, using variously-shaped boxes, you'd quickly run out of shapes. Besides, you know from experience that "A" stands for "area", "h" stands for "height", and "a" and "b" stand for the lengths of the parallel top and bottom sides. Heaven only knows what a square box or a triangular box might stand for!
In other words, they switched from boxes to variables because, while the boxes and the letters mean the exact same thing (namely, a slot waiting to be filled with a value), variables are better. Variables are more flexible, easier to read, and can give you more information.
The same is true of "y" and "f (x)" (pronounced as "eff-of-eks"). For functions, the two notations mean the exact same thing, but "f (x)" gives you more flexibility and more information. You used to say "y = 2x + 3; solve for y when x = −1". Now you say "f (x) = 2x + 3; find f (−1)" (pronounced as "f-of-x equals 2x plus three; find f-of-negative-one"). In either notation, you do exactly the same thing: you plug −1 in for x, multiply by the 2, and then add in the 3, simplifying to get a final value of +1.
But function notation gives you greater flexibility than using just "y" for every formula. For instance, your graphing calculator will list different functions as y1, y2, etc, so you can tell the equations apart when, say, you're looking at their values in table.
In the same way, in textbooks and when writing things out, we use different function names like f (x), g(x), h(x), s(t), etc, to keep track of, and work with, more than one formula in any single context. With function notation, we can now use more than one function at a time without confusing ourselves or mixing up the formulas, leaving ourselves wondering "Okay, which 'y' is this one?" And the notation itself can be usefully explanatory.
From geometry, you know that "A(r) = πr2" indicates the area of a circle, given in terms of the value of the radius r, while "C(r) = 2πr" indicates the circumference of the circle, given in terms of the radius r. Both functions have the same plug-in variable (namely, the "r" variable), but "A" reminds you that the first function is the formula for "area" and "C" reminds you that the second function is the formula for "circumference".
What is the difference between f (x) and y?
There is no difference between "f (x)" and "y". The notation "f (x)" means exactly the same thing as "y". You can even label the y-axis on your graphs with "f (x)", if you feel like it. It doesn't matter if you're graphing y=, looking at Y1= in your calculator, or plugging x-values into f(x)=; they all mean the same thing!
Let me clarify another point. While parentheses have, up until now, always indicated multiplication, that is not the case with function notation. Contrary to all previous experience, the parentheses for function notation do not indicate multiplication.
The expression "f (x)" means "a formula, named f, which has x as its input variable". It does not mean "multiply f and x"!
Don't embarrass yourself by pronouncing (or thinking of) "f (x)" as being "f times x", and never try to "multiply" the function name with its parenthesised input.
In function notation, the "x" in "f (x)" is called "the argument of the function", or just "the argument". So if they give you the expression "f (2)" and ask for the "argument", the answer is just "2".
Why is the input to a function called the "argument"?
The term "argument" has a long history. Originally, it was a logical term, referring to a statement that forwarded a proof or, in a less formal sense, an "argument" was a claim or basic assumption that was being used to try to convince somebody of something. Eventually, the term came to refer, in early scientific contexts, to any mathematical value that was needed as an input to other computations, or any value upon which later results depended.
In the twentieth century, when computer coding started becoming a thing, coders adopted the mathematical meaning to refer to inputs to their coding. In our mathematical context, the "argument" is the independent variable (the one for which you pick a value, usually being the x-value) and the function's output is the dependent variable (the one whose value depends upon whatever was plugged in, usually being the y-value).
- Given h(s), what is the function name, and what is the argument?
I'll do the second part first. The argument is whatever is inside the parentheses, so the argument here is s.
The function name is the variable that comes before the parentheses. In this case, then, the function name is h.
function name: h
argument: s
- What is the argument of f (y)?
The argument is whatever is plugged in. In this particular (unusual) case, the variable being plugged in is "y". (To be fair, there's no rule saying that y can't be the independent variable.) So:
the argument is y
- Given g(t) = t 2 + t, what is the function name? In g(−1), what is the argument?
The function name is what comes before the parentheses, so the function name here is g.
In the second part of the question, they're asking me for the argument. In the first part, where they gave me the function name and argument (being the "g(t)" part) and the formula (being the "t 2 + t" part), the argument was t. But in the second part, they've plugged a particular value in for t. So, in the second part, the argument is the number −1.
function name: g
argument of g(−1): −1
How is a function evaluated at a number?
You evaluate "f (x)" in exactly the same way that you've always evaluated "y"; namely, you take the number they give you for the input variable, you plug it in for the variable, and you simplify to get the answer. For instance:
- Given f (x) = x2 + 2x − 1, find f (2).
To evaluate f (x) at x = 2, I'll plug 2 in for every instance of x in the function's rule:
f (2) = (2)2 +2(2) − 1
To keep things straight in my head (and to keep my logic clear in my working), I've put parentheses around every instance of the argument 2 in the formula for f. Now I can simplify:
(2)2 + 2(2) − 1
= 4 + 4 − 1
= 7
Then my answer is:
f (2) = 7
- Given f (x) = x2 + 2x − 1, find f (−3).
To evaluate, I do what I've always done. I'll plug the given value (−3) in for the specified variable (x) in the given formula:
f (−3) = (−3)2 + 2(−3) − 1
Once again, I've used parentheses to clearly designate the value being input into the formula. In this case, the parentheses are helping me keep track of the "minus" signs. Now I can simplify:
(−3)2 + 2(−3) − 1
= 9 − 6 − 1
= 2
Then my answer is:
f (−3) = 2
If you experience difficulties when working with negatives, try using parentheses as I did above. Doing so helps keep track of things like whether or not the exponent is on the "minus" sign. And it's just generally a good habit to develop.
An important type of function is called a "piecewise" function, so called because, well, it's in pieces. For instance, the following is a piecewise function:
As you can see, this function is split into two halves: the half that comes before x = 1, and the half that goes from x = 1 to infinity. Which half of the function you use depends on what the value of x is. Let's examine this:
- Given the function f (x) as defined above, evaluate the function at the following values: x = −1, x = 3, and x = 1.
This function comes in pieces; hence, the name "piecewise" function. When I evaluate it at various x-values, I have to be careful to plug the argument into the correct piece of the function.
They first want me to evaluate at x = −1. Since this is less than 1, then this argument goes into the first piece of the function. To refresh, the function is this:
Then I'll be plugging the −1 into the rule 2x2 − 1:
f (−1) = 2(−1)2 − 1
= 2(1) − 1
= 2 − 1 = 1
Next, they want me to find the value of f (3). Since 3 is greater than 1, then I'll need to plug into the second piece of the function, so:
f (3) = (3) + 4 = 7
Finally, they want me to evaluate f (x) at x = 1. This is the only x-value that's a little tricky. Which half do I use?
Looking carefully at the rules for the functions, I can see that the first piece is the rule for x-values that are strictly less than 1; the rule does not apply when x equals 1. On the other hand, the second piece applies when x is greater than or equal to 1. Since I'm dealing here with x = 1, then the second piece's rule applies.
f (1) = (1) + 4 = 5
Then my answer is:
f (−1) = 1
f (3) = 7
f (1) = 5
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