People do often say "the function $f(x)$". That's a bad idea for several reasons, but an example illlustrates one of the good ones. Let $U$ denote the set of all function from the reals to the reals. I'm going to define a function $$ g : \Bbb Z \to U $$ by saying that $g(n)$ is the function that takes $x$ to $x^n$. I suppose that I could even write $$ g(n)(x) = x^n, $$ or, to use the definition that some folks like --- "a function is a triple $(D, C, R)$, where $R$ is a subset of $D \times C$ such that ... " --- I could say that $$ g(n) = (\Bbb R, \Bbb R, \{(x, x^n) \mid x \in \Bbb R\}). $$ Show The point here though is that for any number $n$ --- say $n = 2$, the object denoted by $g(2)$ is a particular function -- in this case the "squaring function". So when you say "the function $g(n)$", are you referring to the thing that takes integers to element of $U$, or are you referring to the $n$th-power function? I claim that it's the latter, and that if you want to refer to the former, you should say "the function $g$". When you do computer programming, and actually have to give explicit names and types to things, this sort of distinction matters a lot, although I have to say that many of my colleagues are exceptionally sloppy in the way they describe functions (unless they're actually programming, where the programming language may force them to be precise). If you think my example is contrived, let me give another. Let $C$ be the set of all everywhere-differentiable functions from the reals to the reals. Then I can define a function $$ H: C \to U : f \mapsto f' $$ i.e., for any differentiable function $f$, there's a new function $H(f)$, which is the derivative of $f$. [Not surprisingly, $H(f)$ is often written with some notation involving the letter "d", but I wanted to stay out of that quagmire.] The function $H$ comes up all the time. And now what do you mean when you speak of "the function $H(f)$? Are you referring to the derivative of some particular function $f$, or are you referring the function $H$ itself? Both are objects of interest, and it really helps to have one way to refer to each. If you encounter someone who insists that the function is called $f(x)$ rather than $f$, ask them if $f(y)$ is also a function, and whether it's the same function. [Most reasonable people should say that it is a function, and once you admit that, you kinda have to say it's the same one...] You can then ask whether $f(x) - f(y)$ is in fact zero, because the two things are "the same". At this point, they'll get annoyed with you and say things like "You know what I mean! Don't play the goat!" I recommend walking away, mumbling quietly to yourself "f of x minus f of y...should be zero...hmmm..." A function may be thought of as a rule which takes each member x of a set and assigns, or maps it to the same value y known at its image. x → Function → y A letter such as f, g or h is often used to stand for a function. The Function which squares a number and adds on a 3, can be written as f(x) = x2+ 5. The same notion may also be used to show how a function affects particular values. Example f(4) = 42 + 5 =21, f(-10) = (-10)2 +5 = 105 or alternatively f: x → x2 + 5. The phrase "y is a function of x" means that the value of y depends upon the value of x, so:
Example If f(x) = 3x + 4, find f(5) and f(x + 1). f(5) = 3(5) + 4 = 19 Domain and Range The domain of a function is the set of values which you are allowed to put into the function (so all of the values that x can take). The range of the function is the set of all values that the function can take, in other words all of the possible values of y when y = f(x). So if y = x2, we can choose the domain to be all of the real numbers. The range is all of the real numbers greater than (or equal to) zero, since if y = x2, y cannot be negative. One-to-One We say that a function is one-to-one if, for every point y in the range of the function, there is only one value of x such that y = f(x). f(x) = x2 is not one to one because, for example, there are two values of x such that f(x) = 4 (namely –2 and 2). On a graph, a function is one to one if any horizontal line cuts the graph only once. Composing Functions fg means carry out function g, then function f. Sometimes, fg is written as fog Example If f(x) = x2 and g(x) = x – 1 then As you can see, fg does not necessarily equal gf The Inverse of a Function The inverse of a function is the function which reverses the effect of the original function. For example the inverse of y = 2x is y = ½ x . Example Find the inverse of f(x) = 2x + 1 f-1(x) is the standard notation for the inverse of f(x). The inverse is said to exist if and only there is a function f-1 with ff-1(x) = f-1f(x) = x Note that the graph of f-1 will be the reflection of f in the line y = x. This video explains more about the inverse of a function Graphs Functions can be graphed. A function is continuous if its graph has no breaks in it. An example of a discontinuous graph is y = 1/x, since the graph cannot be drawn without taking your pencil off the paper:  A function is periodic if its graph repeats itself at regular intervals, this interval being known as the period. A function is even if it is unchanged when x is replaced by -x . The graph of such a function will be symmetrical in the y-axis. Even functions which are polynomials have even degrees (e.g. y = x²). The Modulus Function The modulus of a number is the magnitude of that number. For example, the modulus of -1 ( |-1| ) is 1. The modulus of x, |x|, is x for values of x which are positive and -x for values of x which are negative. So the graph of y = |x| is y = x for all positive values of x and y = -x for all negative values of x: Transforming Graphs If y = f(x), the graph of y = f(x) + c (where c is a constant) will be the graph of y = f(x) shifted c units upwards (in the direction of the y-axis). Example The graph of y = |x - 1| would be the same as the above graph, but shifted one unit to the right (so the point of the V will hit the x-axis at 1 rather than 0). What type of function is f/x )= x?The identity function has the same domain and range. The identity function equation is f(x) = x, or y = x.
What are f/x functions called?Algebraic Function:
A function defined by an algebraic expression is called an algebraic function. e.g. f(x)=x2+3x+6.
What are the 4 types of functions in math?Types of Functions. One – one function (Injective function). Many – one function.. Onto – function (Surjective Function). Into – function.. Polynomial function.. Linear Function.. Identical Function.. Quadratic Function.. |
F(x) = x is which function
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