Question

Understanding and drawing graphs of basic functionsand checking the function as one one and onto with respect to their domain and range using graph of the functions.i)x,|x|+1,|x|-1,|x+1|,|x-1| etc

Answer 1

Answer:

Functions and their graphs

Given two sets X and Y, a function from X to Y is a rule, or law, that associates to every element x ∈ X (the independent variable) an element y ∈ Y (the dependent variable). It is usually symbolized as

y = f (x),

in which x is called argument (input) of the function f and y is the image (output) of x under f.

Functions and their graphs

A single output is associated to each input, as different input can generate the same output.

The set X is called domain of the function f (dom f), while Y is called codomain (cod f). The range (or image) of X, is the set of all images of elements of X (rng ƒ). Obviously

rng~f \subseteq cod~f

The notation

f : X \to Y

indicates that ƒ is a function with domain X and codomain Y.

Functions and their graphs

Given ƒ:X → Y, the graph G( f ) is the set of the ordered pairs

(x,y) ~such~ that ~x \in X, y = f (x)

In particular, if x and y are real numbers, G(f ) can be represented on a Cartesian plane to form a curve. A glance at the graphical representation of a function allows us to visualize the behaviour and characteristics of a function.

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Answer 2

Given:

A set of functions x, |x|+1, |x|-1, |x+1|, |x-1|.

To Find:

Whether the given functions are one-one or onto.

Solution:

The given problem can be solved by using the concepts of functions.

1. A function is said to be one-one when every element in the domain has only a single value of the range. At most 1 value is allowed, if a function has two solutions it is not considered as a one-one function.

2. A function is said to be onto if every value in the range is covered in their respective intervals. A function can be both one-one and onto at the same time.

3. The domain of a modulus function is always positive real numbers and generally modulous functions are not one-one in many cases.

3. Consider the first function x,

• The range of the given function is from (-infinite,+infinite),
• The domain of the given function is Real number set,
• Every value of x has a single value and all the values in the range are occupied.
• Hence the given function is both one-one and onto.

4. |x|+1

• The range of the given function is   (-infinite,+infinite),
• The domain of the given function if Positive real numbers,
• The given function is not one-one as the graph intersects at two points. It is an onto function as all the points in the range is covered.

5. |x| - 1

• The range of the given function is   (-infinite,+infinite),
• The domain of the given function if Positive real numbers,
• The given function is not one-one as the graph intersects at two points. It is an onto function as all the points in the range is covered.

6. |x+1|

• The range of the given function is   (-infinite,+infinite),
• The domain of the given function if Positive real numbers,
• The given function is not one-one as the graph intersects at two points. It is an onto function as all the points in the range is covered.

7. |x-1|

• The range of the given function is   (-infinite,+infinite),
• The domain of the given function if Positive real numbers,
• The given function is not one-one as the graph intersects at two points. It is an onto function as all the points in the range is covered.