`star` Functions

`star` Some functions and their graphs

`star` Algebra of real functions

`star` Some functions and their graphs

`star` Algebra of real functions

● We study a special type of relation called function. It is one of the most important concepts in mathematics.

● We can, visualise a function as a rule, which produces new elements out of some given elements. There are many terms such as ‘map’ or ‘mapping’ used to denote a function.

`\color{blue} ☛\color{blue} ul("Function") `

A relation `f` from a set `A` to a set `B` is said to be a function if every element of set `A` has one and only one image in set `B`.

`\color{green} ✍️` In other words, a function `f` is a relation from a non-empty set `A` to a non-empty set `B` such that the domain of `f` is `A` and no two distinct ordered pairs in `f` have the same first element.

`\color{green} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`

`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`

`\color{blue} ☛ color{blue} ul ("Real valued function")`

A function which has either `R` or one of its subsets as its range is called a real valued function. Further, if its domain is also either `R` or a subset of `R`, it is called a real function.

● We can, visualise a function as a rule, which produces new elements out of some given elements. There are many terms such as ‘map’ or ‘mapping’ used to denote a function.

`\color{blue} ☛\color{blue} ul("Function") `

A relation `f` from a set `A` to a set `B` is said to be a function if every element of set `A` has one and only one image in set `B`.

`\color{green} ✍️` In other words, a function `f` is a relation from a non-empty set `A` to a non-empty set `B` such that the domain of `f` is `A` and no two distinct ordered pairs in `f` have the same first element.

`\color{green} ✍️ ` If `f` is a function from `A` to `B` and `(a, b) ∈ f,` then `f (a) = b,` where `b` is called the image of a under `f` and `a` is called the preimage of `b` under `f.`

`\color{green} ✍️ ` The function `f` from `A` to `B` is denoted by `f: A -> B.`

`\color{blue} ☛ color{blue} ul ("Real valued function")`

A function which has either `R` or one of its subsets as its range is called a real valued function. Further, if its domain is also either `R` or a subset of `R`, it is called a real function.

Q 3017291189

Examine each of the following relations given below and state in each case, giving reasons whether it is a function or not?

(i) `R = {(2,1),(3,1), (4,2)}, (ii) R = {(2,2),(2,4),(3,3), (4,4)}`

(iii) `R = {(1,2),(2,3),(3,4), (4,5), (5,6), (6,7)}`

(i) `R = {(2,1),(3,1), (4,2)}, (ii) R = {(2,2),(2,4),(3,3), (4,4)}`

(iii) `R = {(1,2),(2,3),(3,4), (4,5), (5,6), (6,7)}`

(i) Since 2, 3, 4 are the elements of domain of R having their unique images, this relation R is a function.

(ii) Since the same first element 2 corresponds to two different images 2 and 4, this relation is not a function.

(iii) Since every element has one and only one image, this relation is a function.

Q 3067491385

Let `N` be the set of natural numbers. Define a real valued function `f : N → N` by `f (x) = 2x + 1.` Using this definition, complete the table given below.

x | 1 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|

y | f(1) = ......... | f(2) = ........ | f(3) = ...... | f(4) = ... | f(5) = ..... | f(6) = ...... | f(7) = .... |

The completed table is given by

x | 1 | 3 | 4 | 5 | 6 | 7 | |
---|---|---|---|---|---|---|---|

y | f(1) = 3 | f(2) = 5 | f(3) = 7 | f(4) =11 | f(5) = 13 | f(6) = 15 | f(7) = 15 |

Q 3008401308

Let `f = {(1,1), (2,3), (0, –1), (–1, –3)}` be a linear function from Z into Z. Find f(x).

Since f is a linear function, `f (x) = mx + c`. Also, since `(1, 1), (0, – 1) ∈ R,`

`f (1) = m + c = 1` and `f (0) = c = –1`. This gives `m = 2` and `f(x) = 2x – 1.`

● Let `R` be the set of real numbers.

● Define the real valued function `f : R → R` by `y = f(x) = x` for each `x ∈ R.` Such a function is called the identity function.

● Here the domain and range of `f` are `R.` The graph is a straight line as shown in Fig . It passes through the origin.

● Define the real valued function `f : R → R` by `y = f(x) = x` for each `x ∈ R.` Such a function is called the identity function.

● Here the domain and range of `f` are `R.` The graph is a straight line as shown in Fig . It passes through the origin.

● Define the function `f: R → R` by `y = f (x) = c, x ∈ R` where `c` is a constant and each `x ∈ R.`

● Here domain of `f` is `R` and its range is `{c}.`

● The graph is a line parallel to `x`-axis. For example, if `f(x)=3` for each `x∈R,` then its graph will be a line as shown in the Fig .

● Here domain of `f` is `R` and its range is `{c}.`

● The graph is a line parallel to `x`-axis. For example, if `f(x)=3` for each `x∈R,` then its graph will be a line as shown in the Fig .

● A function `f : R→R` is said to be polynomial function if for each `x` in `R, `

● `y = f (x) = a_0 + a_1x + a_2x^2 + ...+ a_n x^n,` where `n` is a non-negative integer and `a_0, a_1, a_2,...,a_n ∈R.`

●The functions defined by `f(x) = x^3 – x^2 + 2,` and `g(x) = x^4 + 2 x` are some examples of polynomial functions,

whereas the function `h` defined by `h(x) = x^(2/3) +2x` is not a polynomial function.

● `y = f (x) = a_0 + a_1x + a_2x^2 + ...+ a_n x^n,` where `n` is a non-negative integer and `a_0, a_1, a_2,...,a_n ∈R.`

●The functions defined by `f(x) = x^3 – x^2 + 2,` and `g(x) = x^4 + 2 x` are some examples of polynomial functions,

whereas the function `h` defined by `h(x) = x^(2/3) +2x` is not a polynomial function.

Q 3017591489

Define the function `f: R → R` by `y = f(x) = x^2, x ∈ R.` Complete the Table given below by using this definition. What is the domain and range of this function? Draw the graph of `f`.

The completed Table is given below:

Domain of `f = {x : x∈R}`. Range of `f = {x : x ≥ 0, x ∈ R}.` The graph of `f` is given by Fig

Q 3057691584

Draw the graph of the function `f : R → R` defined by `f (x) = x^3, x ∈ R.`

We have

`f(0) = 0, f(1) = 1, f(–1) = –1, f(2) = 8, f(–2) = –8, f(3) = 27; f(–3) = –27,` etc.

Therefore, `f = {(x,x^3): x∈R}`.

The graph of `f` is given in fig

● Rational functions are functions of the type `(f( x))/ (g (x))` ,

where `f(x)` and `g(x)` are polynomial functions of `x` defined in a domain, where `g(x) ≠ 0.`

where `f(x)` and `g(x)` are polynomial functions of `x` defined in a domain, where `g(x) ≠ 0.`

Q 3028201101

Define the real valued function `f : R – {0} → R` defined by `f (x) = 1 /x , x ∈ R –{0}.` Complete the Table given below using this definition. What is the domain and range of this function?

The completed Table is given by The domain is all real numbers except 0 and its range is also all real numbers except 0. The graph of f is given in

● The function `f: R→R` defined by `f(x) = |x|` for each `x ∈R` is called modulus function.

● For each non-negative value of `x, f(x)` is equal to `x.` But for negative values of `x`, the value of `f(x)` is the negative of the value of `x,` i.e.,

`f(x)= { tt (( x , if , x >= 0),(-x, if, x< 0))`

The graph of the modulus function is given in Fig .

● For each non-negative value of `x, f(x)` is equal to `x.` But for negative values of `x`, the value of `f(x)` is the negative of the value of `x,` i.e.,

`f(x)= { tt (( x , if , x >= 0),(-x, if, x< 0))`

The graph of the modulus function is given in Fig .

● Signum function The function `f:R→R` defined by

`f(x)= { tt (( 1, if , x > 0),(0, if, x=0),( -1, if , x < 0))`

is called the signum function.

● The domain of the signum function is R and the range is the set `{–1, 0, 1}`.

● The graph of the signum function is given by the Fig .

`f(x)= { tt (( 1, if , x > 0),(0, if, x=0),( -1, if , x < 0))`

is called the signum function.

● The domain of the signum function is R and the range is the set `{–1, 0, 1}`.

● The graph of the signum function is given by the Fig .

● The function `f: R → R` defined by

● `f(x) = [x], x ∈R` assumes the value of the greatest integer, less than or equal to `x.` Such a function is called the greatest integer function.

From the definition of `[x],` we can see that

`tt (( [x]= -1, "for" \ \ –1 ≤ x < 0 ),([x]=0 , "for" \ \ 0 ≤ x < 1 ),( [x] = 1 , "for" \ \ 1 ≤ x < 2 ),([x] = 2, "for" \ \ 2 ≤ x < 3))`

and so on.

The graph of the function is shown in Fig .

● `f(x) = [x], x ∈R` assumes the value of the greatest integer, less than or equal to `x.` Such a function is called the greatest integer function.

From the definition of `[x],` we can see that

`tt (( [x]= -1, "for" \ \ –1 ≤ x < 0 ),([x]=0 , "for" \ \ 0 ≤ x < 1 ),( [x] = 1 , "for" \ \ 1 ≤ x < 2 ),([x] = 2, "for" \ \ 2 ≤ x < 3))`

and so on.

The graph of the function is shown in Fig .

`(i) "Addition of two real functions"` Let `f : X → R` and `g : X → R` be any two real functions, where `X ⊂ R.`

Then, we define `(f + g): X → R` by

●`(f + g) (x) = f (x) + g (x),` for all `x ∈ X.`

`(ii) "Subtraction of a real function from another"` Let `f :X → R` and `g:X → R` be any two real functions, where `X ⊂ R.`

Then, we define `(f – g) : X→R` by

● `(f–g) (x) = f(x) –g(x),` for all `x ∈ X.`

`(iii) "Multiplication by a scalar"` Let `f : X→R` be a real valued function and `α` be a scalar. Here by scalar, we mean a real number.

Then the product `α f` is a function from `X` to `R` defined by

● `(α f ) (x) = α f (x), x ∈X.`

`(iv) "Multiplication of two real functions"` The product (or multiplication) of two real functions `f:X→R` and `g:X→R` is a function

● `fg:X→R` defined by `(fg) (x) = f(x) g(x),` for all `x ∈ X.`

● This is also called `"pointwise multiplication."`

`(v) "Quotient of two real functions"` Let `f` and `g` be two real functions defined from `X→R` where `X ⊂R.`

The quotient of `f` by `g` denoted by `f/g` is a function defined by ,

● `(f/g)(x)=(f(x))/(g(x))` provided `g(x) ≠ 0, x ∈ X`

Then, we define `(f + g): X → R` by

●`(f + g) (x) = f (x) + g (x),` for all `x ∈ X.`

`(ii) "Subtraction of a real function from another"` Let `f :X → R` and `g:X → R` be any two real functions, where `X ⊂ R.`

Then, we define `(f – g) : X→R` by

● `(f–g) (x) = f(x) –g(x),` for all `x ∈ X.`

`(iii) "Multiplication by a scalar"` Let `f : X→R` be a real valued function and `α` be a scalar. Here by scalar, we mean a real number.

Then the product `α f` is a function from `X` to `R` defined by

● `(α f ) (x) = α f (x), x ∈X.`

`(iv) "Multiplication of two real functions"` The product (or multiplication) of two real functions `f:X→R` and `g:X→R` is a function

● `fg:X→R` defined by `(fg) (x) = f(x) g(x),` for all `x ∈ X.`

● This is also called `"pointwise multiplication."`

`(v) "Quotient of two real functions"` Let `f` and `g` be two real functions defined from `X→R` where `X ⊂R.`

The quotient of `f` by `g` denoted by `f/g` is a function defined by ,

● `(f/g)(x)=(f(x))/(g(x))` provided `g(x) ≠ 0, x ∈ X`

Q 3058201104

Let `f(x) = x^2` and `g(x) = 2x + 1` be two real functions.Find `(f + g) (x), (f –g) (x), (fg) (x), (f/g) (x)`

We have,

`(f + g) (x) = x^2 + 2x + 1, (f –g) (x) = x^2 – 2x – 1,`

`(fg)(x) = x^2 (2x+1) = 2x^3 + x^2 , (f/g) (x) = x^2/(2x+1) , x ne -1/2`

Q 3038301202

Let `f(x) = x` and `g(x) = x` be two functions defined over the set of non negative real numbers. Find `(f+g) (x) , (f-g) (x) , (fg) (x)` and `(f/g) (x)`

We have

`(f + g) (x) = sqrtx + x, (f – g) (x) = sqrtx-x`

`(fg) x = sqrtx (x) = x^(3/2)` and `(f/g) (x) = sqrtx/x = x^(-1/2) , x ne 0`