What Does F 2 Mean
Function Notation & Evaluating at Numbers
Y'all've been playing with " y=" sorts of equations for some fourth dimension at present. And you've seen that the "squeamish" equations (direct lines, say, rather than ellipses) are the ones that you can solve for " y=" and and so plug into your graphing computer. These " y=" equations are functions. But the question y'all are facing at the moment is "Why do I demand this function notation, particularly when I've got perfectly nice ' y=', and how does this notation piece of work?"
Remember back to when you lot were in elementary schoolhouse. Your instructor gave y'all worksheets containing statements like "[ ] + ii = 4" and told you to fill in the box. In one case yous got older, your teacher started giving you worksheets containing statements like " ten + 2 = iv" 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 expanse A of a trapezoid with upper base a , lower base b , and meridian h ? The formula is as follows:
If you endeavour to express the above, or something more than complicated, using variously-shaped boxes, you'd chop-chop run out of shapes. Besides, you know from experience that " A " stands for "surface area", " h " stands for "elevation", and " a " and " b " represent the lengths of the parallel top and bottom sides. Heaven only knows what a foursquare box or a triangular box might represent!
In other words, they switched from boxes to variables considering, while the boxes and the letters mean the exact same matter (namely, a slot waiting to be filled with a value), variables are ameliorate. Variables are more flexible, easier to read, and tin 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 verbal same thing, just " f(x)" gives you more than flexibility and more than data. Yous used to say " y = 210 + 3; solve for y when x = −1". Now you say " f(10) = 2x + three; find f(−1)" (pronounced equally " f -of- x equals 2ten plus three; observe f -of-negative-1"). In either notation, you exercise exactly the aforementioned affair: yous plug −1 in for x , multiply by the ii, and and then add in the 3, simplifying to become a terminal value of +1.
But part notation gives you greater flexibility than using merely " y " for every formula. For instance, your graphing calculator will list different functions as y1, y2, etc, so you tin can tell the equations apart when, say, you're looking at their values in "Table".
In the same style, in textbooks and when writing things out, we apply different function names similar f(x), g(x), h(x), south(t), etc, to continue runway of, and work with, more one formula in whatsoever single context. With function notation, nosotros can at present use more than one function at a time without confusing ourselves or mixing upwards the formulas, leaving ourselves wondering "Okay, which ' y ' is this ane?" And the notation can be usefully explanatory.
From geometry, you lot know that " A(r) = πr two " indicates the area of a circle given in terms of the value of the radius r , while " C(r) = 2πr " indicates the circumference given in terms of the radius r . Both functions take the same plug-in variable (the " r "), simply " A " reminds you that the first part is the formula for "expanse" and " C " reminds you that the 2nd function is the formula for "circumference".
Think: The notation " f(10)" is exactly the same matter equally " y ". You can even label the y -axis on your graphs with " f(x)", if you experience like it.
Let me clarify some other point. While parentheses accept, up until at present, always indicated multiplication, that is not the case with role notation. Contrary to all previous experience, the parentheses for office note do not bespeak multiplication.
The expression " f(ten)" ways "a formula, named f , has x every bit its input variable". It does not mean "multiply f and x "!
Don't embarrass yourself past pronouncing (or thinking of) " f(x)" as being " f times ten ", 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 y'all the expression " f(2)" and ask for the "argument", the answer is only "ii".
Why is the input to a function called the "argument"?
The term "statement" has a long history. Originally, it was a logical term, referring to a argument that forwarded a proof or, in a less formal sense, a claim that was existence 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 calculator coding started becoming a thing, coders adopted the mathematical significant to refer to inputs to their coding. In our mathematical context, the "argument" is the independent variable (the one for which you choice a value, usually beingness the x -value) and the function's output is the dependent variable (the i whose value depends upon whatever was plugged in, usually beingness the y -value).
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Given h(southward), what is the office name, and what is the statement?
I'll do the 2d role first. The argument is whatever is inside the parentheses, and so the argument hither is s .
The office name is the variable that comes before the parentheses. In this case, then, the function name is h .
office name: h
argument: s
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What is the argument of f(y)?
The statement is whatever is plugged in. In this detail (unusual) example, the variable being plugged in is " y ". (Later on all, there'south no rule saying that y can't be the independent variable.) Then:
the argument is y
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Given g(t) = t two + t , what is the function name? In thou(−one), what is the argument?
The function name is what comes before the parentheses, so the function proper noun here is one thousand .
In the second part of the question, they're asking me for the argument. In the first function, where they gave me the function name and argument (existence the " grand(t)" part) and the formula (being the " t two + t " role), the argument was t . Simply in the 2nd function, they've plugged a particular value in for t . Then, in the second part, the argument is the number −1.
function name: g
argument of k(−i): −1
How is a role evaluated at a number?
You evaluate " f(x)" in exactly the same way that you've always evaluated " y "; namely, you lot accept the number they give y'all for the input variable, you plug information technology in for the variable, and you simplify to get the answer. For example:
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Given f(x) = x 2 + 2ten − i, find f(2).
To evaluate f(ten) at 10 = 2, I'll plug ii in for every instance of ten in the function'southward rule:
f(two) = (2)2 +2(2) − 1
To keep things direct in my caput (and articulate in my working), I've put parentheses effectually every instance of the argument two in the formula for f . Now I tin can simplify:
Then my answer is:
f(2) = 7
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Given f(x) = x 2 + 2x − 1, find f(−3).
To evaluate, I practice what I've e'er done. I'll plug the given value (−3) in for the specified variable ( x ) in the given formula:
f(−iii) = (−iii)2 + 2(−3) − i
Once again, I've used parentheses to clearly designate the value being input into the formula. In this case, the parentheses are helping me continue track of the "minus" signs. At present I tin simplify:
Then my answer is:
f(−3) = 2
If yous experience difficulties when working with negatives, attempt using parentheses as I did to a higher place. Doing so helps keep runway of things like whether or not the exponent is on the "minus" sign. And it'southward just generally a good habit to develop.
An important type of function is called a "piecewise" function, and then called because, well, it's in pieces. For case, the post-obit is a piecewise part:
As yous can see, this function is split into two halves: the half that comes earlier ten = 1, and the half that goes from x = 1 to infinity. Which half of the part yous use depends on what the value of x is. Allow'southward examine this:
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Given the function f(x) as defined to a higher place, evaluate the function at the following values: ten = −1, x = three, and ten = ane.
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 one, and so this argument goes into the first piece of the function. To refresh, the role is this:
Then I'll exist plugging the −i into the rule 210 2 − 1:
Next, they want me to find the value of f(iii). Since three is greater than 1, so I'll need to plug into the 2d piece of the role, and then:
f(3) = (3) + four = vii
Finally, they desire me to evaluate f(x) at x = i. This is the simply 10 -value that's a piddling tricky. Which half exercise I use?
Looking carefully at the rules for the functions, I can run across that the first piece is the rule for x -values that are strictly less than i; the dominion does not apply when x equals 1. On the other hand, the second slice applies when x is greater than or equal to 1. Since I'm dealing here with ten = 1, and then the second piece's rule applies.
f(1) = (1) + 4 = 5
Then my reply is:
f(−1) = 1
f(iii) = vii
f(1) = 5
What Does F 2 Mean,
Source: https://www.purplemath.com/modules/fcnnot.htm
Posted by: sosakinge1950.blogspot.com
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