Mathematics 1 Dersi 2. Ünite Sorularla Öğrenelim

What is a function?

A correspondence assigning each element of the set A, one and only one element of the set B
is called a function from the set A into the set B. Functions are generally denoted by the lower case letters such as f, g, h. In this case a function defined from the set A to the set B is denoted by f : A › B.

For the function f : A › B, what is the domain?

the set A is called the domain or the departure set of the function.

For f : A › B, what is the range?

the set B is called the range or terminal set.

What is the image set of the function?

The set of all elements composed of the images of each element of A is called the image set of the function.

What is natural domain?

If a rule defining a function is given but the domain has not been specified explicitly, then
the largest set which makes the rule meaningful is understood. This set is denoted by Df
for the rule y=f (x) and is called the natural domain of the function.

What is an one-to-one function?

If the images of two different elements from the domain under the action of f are also different, such functions are called one-to-one.

What are surjective functions?

Although the range and the image of a function are two concepts that need to be distinguished, we still know that they are closely related. The image is always a subset of the range. Yet, there are functions for which the range and the image sets are exactly the same. Thus, given a function f : A › B if the image is equal to the range, i.e. f (A)=B the function f is called surjective (onto). Equivalently, for every element b of the set B if there can be found an element a of A such that f (a)=b, then f is called onto.

What is a Bijective Function?

If a function which is both one-to-one and onto is called a bijection.

What is a Constant Function?

A function assigning each element from its domain a single element in its range is called a
constant function. That is, for every element a from the domain A, and c?B, if f (a)=c, the
function f is called constant.

What is an Identity Function?

Given A ?ø, a function defined on the set A and assigning every element of A to itself is called
the identity function.

If the elements of sets have a common property, what can be done?

Just as sets have different representations, functions also have different representations. If
sets have a small number of elements, we may list them. If the elements of sets have a common
property, we denote them by emphasizing this common property.

What are piecewise defined functions?

Functions, which are represented by different formulas on different subsets of its domain are called piecewise defined functions. We frequently encounter piecewise defined functions in our daily lives. For example, suppose that you will be parking your car in a car park. The car park charges 5 T for the first hour. For every following hour, it charges an extra 1,5 T. How much would you pay if you pick your car during the first hour? How much would you pay 1 hour 40 minutes later? How about 2 hours 15 minutes later?
Here, we have a piecewise defined function. We may define this function in the following way: If x
corresponds to the parking time in hours, we have
f :R+ › R

What is an absolute value function?

A function which assigns every real number to its distance to the origin, in other words assigning
its absolute value, is called the absolute value function, and it is denoted by | • |. As a piecewise defined function, absolute value is represented as
l·l :R › R, x ? R

What is the composition of the functions f and g of  f : A›B and g : B›C?

There are many ways of constructing new functions from the given ones. The most important way of constructing a new function from the known ones is the composition of them. We now give the definition of composition. Let the functions f : A›B and g : B›C be given.
The function g°f : A›C, defined by the rule (g°f )(a)=g(f (a)) is called the composition of the functions f and g.

What is the fuction called when  the sets {a, b, c, d } and {1, 2, 3, 4}are both  one-to-one and onto function?

The matching of the sets {a, b, c, d } and {1, 2, 3, 4} thus obtained is a function which we call as the inverse function of h. Consequently, having a oneto-one and onto function, it is possible to define a new function by reversing the directions of the arrows as we did in the preceding examples. This newly constructed function is called the inverse of the given function. Mathematically put: Let the bijective function f : A›B be given. The inverse of f is defined as
f –1:B›A, f –1(y)=x

What kind of operations can be done with functions?

Just as the operations of addition, subtraction, multiplication, and division defined on real numbers, it is possible to define similar operations on functions defined from a subset A of real
numbers to real numbers, or a subset of it.

What is the Cartesian coordinate system?

Cartesian coordinate system, a tool which enables us to view the graphs of functions.

What is the horizontal real line called in the Cartesian coordinate system?

The horizontal real line is called the x-axis.

What is the vertical real line called in Cartesian coordinate system?

The vertical real line is called the y-axis, or ordinate.

What is the graph of a function?

The set of ordered pairs composed of (x, f (x)) for every element x in A is called the graph of the function f :A? R › R .

What kind of functions have an inverse?

Only bijective functions have an inverse.

What is the largest domain of the function f(x)= 1/x?

The largest domain of the function f(x)= 1/x is the set which makesthe rule meaningfull. f(x)= 1/x
makes sense if its denominator isnot zero, i.e. x ? 0 should be satisfied. Thus, the largest domain Df should be taken as R ›R\{0}.

How can you find the composition of two functions?

Let the functions f :A›B and g :B›C be given. You can construct the composition of the functions f and g as g °f :A›C, (g °f )(x)=g (f (x)), for x ?A.