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FUNCTIONS - Coggle Diagram
FUNCTIONS
GRAPHS OF FUNCTIONS
GRAPHS OF LINEAR FUNCTIONS
GRAPHS OF PIECEWISE FUNCTIONS
GRAPHS OF ABSOLUTE VALUE
|f(x)|={f(x), f(x) ≥ 0 , {-f(x), f(x)<0
If the elements of a function come in corresponding, those that are the set of points on the analytic plane are called the function graph. It is symbolically represented as: The set f = {(x, y): y = f (x), (x, y) ∈AxB} given the function f:A ----> B is a graph with a function.
EVEN AND ODD FUNCTIONS
The even function is symmetric with respect to the y axis.
A odd function is symmetric with respect to the origin.
f:R---->R for ∀x∈ even only exponent of functions have even numbers. The exponent of odd numbers is just an odd number. If f(-x)=f(x), f will be even function. If f(-x)=-f(x), f will be odd function.
ONE TO ONE FUNCTIONS
f:A--->B if the image of each element is different in domain, it is called a one-to-one function and it is represented as f(a)≠f(b). If it is become f(a)=f(b), this will be a=b. Then f becomes one to one functions. It is denoted 1-1.
INTO FUNCTIONS
f:A--->B If the number of elements of the image set is less than the number of elements of the value set, it is called into function. It is represented as f(A)≠B.
ONTO FUNCTIONS
When the set of images is equal to the set of values, it is called a onto function.
f:A--->B, f(A)=B ---> f is become onto functions also f:A-->B if it is becomes a∈A, for all b∈B, example of that b=f(a) f becomes onto functions
IDENTITY FUNCTIONS
If the domain and range are equal and they all go to each other separately, it becomes the identity function.l:A--->A defined by ı(x)=x for all x∈A it is becomes identitiy functions.
LINEAR FUNCTIONS
f:R---->R It doesn't matter a real number a and b, If so (x) = ax + b this is called a linear function and the largest degree term is x.
INVERSE OF FUNCTIONS
If the function in the set f:A--->B is one-to-one and onto functions, it becomes the inverse function and it becomes f:B-->A also it can be show like this f(x)=y, f-1 (y)=x and it denoted as f-1.
(f-1)-1=f , (fof)-1=(f-1of)=I , (fog)-1=g-1o f-1 , (fog)oh=fo(goh)
COMPOSITION OF FUNCTIONS
A, B, and C are non-empty sets. Let the functions be f:A --> B and g:B --> C. The functions gof = {(x, z) | x∈A and z = g (f (x)) C} also it is denoted fog(x)=f(g(x).
If a function with a unit function has a composition, it will be equal to the function itself. Like this foI=f
The combination of functions is associative. It is shown as fo(goh)=(sis)oh.
The function composition is not commutative . Therefore, the fog dose need not be equal to gof. It is denoted fog(x)≠gof(x)
If the functions f:A--->B and g:B--->C are 1-1,the function will be gof:A--->C is also become 1-1.
If the functions f:A--->B and g:B--->C are onto, the function will be gof:A--->C is also become onto.
VERTICAL LINE TEST
To understand whether the graph is a function or not, lines parallel to the y axis are drawn on the graph. Vertical lines become a function if they intersect the graph at only one point. If vertical lines intersect the graph at two or more points, the graph is not a function.
WHAT IS THE MEANING OF THE FUNCTION?
The relation f, which pairs each element of A with only one element of B and it is called a function from A to B.
Each element in the domain is related with an element in the range. Any element in the domain must be related with only one element in the range.
If the function is from A to B, then A ---> B is shown as.
c∈A, d∈B this shown like this f: c--->d, f(c)=d
EQUAL FUNCTIONS
f:A--->B If the domain and range are equal, the image in the range and domain are equal, and the coefficients of the same variables are equal to each other, it is called an equal function. For example, if f (z)=g (z) is equal, it is denoted as f =g.
PIECEWISE FUNCTIONS
The function represented by the rules in the sub-ranges of the field is called a piecewise function. The rule is that discrete subsets of domain are different from each other. It is shows in this way:
f(x)={g(x), x ≤a
a∈R
{r(x), x>a
CONSTANT FUNCTIONS
f:A--->B If all the elements in a go to one element in b, it becomes a constant function. It is denoted f(x)=y ,then y is become constant real number. If f (x) mx + n is a constant function, m equals zero and n is a real number. Fx= ax+b/ cx+d if f is constant function, it becomes a/c= b/d.
FOUR OPERATIONS IN FUNCTIONS
f:A-->R, g:B-->R, (A∩B)≠∅ , f+g: (A∩B)--->R, (f+g)(x)=f(x)+g(x) , f-g:(A∩B)---->R, (f.g)(x)=f(x).g(x) , f/g:(A∩B)--->R, (f/g)(x)= f(x)/g(x) g(x)≠0, d∈R, ∀x∈A, (d.f)(x)=d.f(x)
