Good afternoon students, today, we will study Functions 2. In functions 1, we talked about different

kinds of functions, how to solve inequalities, how to find domain of functions, how to find range of

function. What are one, one functions, what are many to one functions, what are into functions and

what are onto functions? Now, we will continue from there only. First of all we will take few

questions from revisions on domain and range.

Okay, first question is, find the domain of fx equals to log of 4 minus x square cum mod to the base

10. How to solve, we had learnt that whatever is written inside the log, what it should be, greater

than zero. So, mod of 4 minus x square should be greater than zero. Base also should be greater zero

and base should not be 1. Base is already 10, it is greater than zero and it is also not 1. So, only mod

of 4 minus x square should be greater than zero. Mod of anything is always non-negative. So,

modulus of anything cannot be negative. So, when it will be greater than zero, when it is not zero.

Means x square should not be equal to 4, x should not be equal to 2 and x should not be equal to

minus 2. So, in this domain all x will be except 2 and minus 2. Hence, domain is x belongs to all real

numbers except 2 and minus 2 or x belongs to minus infinite to minus 2, union minus 2 to 2 union 2

to infinite. They have not given anything in the option, actually this option is wrong it should be curly

bracket and this becomes correct option. Leave the options, you can see these are the correct form

of answers.

Let us solve next question. Here there is also one small mistake. Here equal to sign should not be

there. We have to find domain, question is find domain. Question is find domain. There are two

functions and both of them should be satisfied. When the first log is satisfied, when the expression

inside is greater than zero, that is x should be greater than half. And the second log is satisfied, when

the expression inside it is greater than zero. Square root of any number is always greater than zero.

What we have to keep in mind is that, whatever quantity is inside square root it should be greater

than zero. And we know it is 2x minus 1 whole square plus 4, 4 plus pause, non-negative number. If a

square is added to 4 it will always be greater than zero, in fact it will be bigger than 4. It is greater

than zero for all x belonging to real numbers. x should be greater than half and should follow this

and what we will take the intersection of both and what are they doing to x, satisfy and those x are

half to infinite. Correct the question, find domain.

Next question, this question is about finding range. This question is about finding range of this

expression. We can see in this expression, one same term is used two times that is x square. We can

write it like this and like this hence what is x square minus y upon y minus 1 and we know that x

square is always greater than or equal to zero which implies minus y upon y minus 1 is always

greater than or equal to zero. Hence, y upon y minus 1 is less than or equal to zero. Wavy curve

method, where is zero, on zero, if we put 1 in denominator, 1 will not come in the answer. So, where

it is negative and zero, hence it is range, zero close, 1 open. Means its value will be in between zero

and 1, it could be zero but not 1. Clear? We found the value of x square and whatever we knew

about x square, that always x square is zero or greater than zero and there only we got minus y upon

y minus 1 and we solved with Wavy Curve. And we can visualise easily, how, x square upon 1 plus x

square, denominator is always greater. Denominator is always big, positive upon positive. So, there

can only be positive number in y, y cannot be negative. So, denominator will be greater, if

denominator is greater means it will be less than 1, so it will not be greater than 1. y will not be

greater than 1, number greater than 1 won’t come in the range. Can 1 come in the range?

Numerator upon denominator when it will become 1 when both are equal. Both of them cannot be

equal because what is always greater, denominator. Hence, 1 is not in range, here 1 is in range.

Hence, answer cannot be these two. From here we can visualise that answer is zero to 1 because

denominator is greater and it is positive, zero to 1, it will be smaller than 1. We can do this by hit and

trial and by observing also. So, we know what is domain, what is range. We have done a quick

revision.

Let us go further, next what we will study is different types of functions and in this different types

we will talk about even and odd functions. Even functions and Odd functions. What is even function?

For example, y is equal to x square. In y is equal to x square, the value of minus 1 square is 1 and as

well the square value of 1 is also 1. So, the value of minus 1 and 1 is same. So, value of minus 2

square is 4 and value of 2 square is 4. Minus 2 and 2 is same. So, that function whose value of x and

minus x is same, that kind of functions we call as even functions. Even and Odd functions, if f of

minus x is equal to fx, then fx is even function. If f of minus x is equal to fx then fx is even function

and if f of minus x is equal to minus fx, then fx is odd function. For example, f.