Function and Polynomials

FUNCTION

Let A and B be two non-empty sets, then, every subset of cartesian product A\(\times \)B is a relation from A to B. Thus, a relation in which every element of set A is related (or associated) with a unique element of set B is said to be a function from A to B. Such as function is denoted by f : A→ B, which is read as "f is a function from A to B".

Symbolically, it follows that f: A → B if;

(i) f ⊆A×B

(ii) for each given x ∈ A, there exist a unique y∈B such that (x, y) ∈ f.

If an element y of B is associated with an element x of A, then y is an image of x under f and x is called the pre-image of y under f. We can also write y=f(x) and read f(x) as "f of x" or "f at x". A function is usually denoted by f, g, F, G, etc. A function is a relation in which, no two different ordered pairs have the same first component.

Composite Function

Let f and g be the function from A to B and from B to C, respectively, where A= {a1,b2,c3}, B = {c3,d4,e5} and C = {e5, f6, g7,h8,i9} and these are defined as follows:

Here, if f : A →B, a1 maps into c3 ; b2 maps into d4 ; and c3 maps into e5.

If g: B→ C, c3 maps into f6 ; d4 maps into g7 ; and e5 maps into h8.

thus we may write c3= f (a1), d4 = f(b2), e5= f(c3) and f6= g(c3), g7 = g(d4), h8 = f(e5)

Then, we have f6 = g(c3) = g [f(a1)],

g7 = g(d4) = g[f(b2)]

and h8=g(e5) = g[f(e3)]

this defines a set of ordered pairs of a function from A into C. This function is called the (product) composite function of g and f. it is denoted by gof or simply by gof. the nation gof indicates that f is applied first and then g. Thus, the arrow diagram of the new function is alongside.

Inverse Function

Let f: A→ B be a function from A to B defined by the following arrow.

thus, the function f = { (a,1), (b,2) (c,3) } is one -one into where the domain of f = {a, b, c} and the range of f= { 1,2,3 }.

If we interchange the domain set of 'f' into range set and range set of into domain set, the set of ordered pairs will become {(1, a), (2, b), (3, c)}. Let this function e g. Then the function.

g= {(1, a), (2, b) (3, c) } is again a one-one onto, where the domain of g= {1,2,3} and the range of g={a,b,c}

the arrow diagram of this function g is as follows.

in such case, the function g is called the inverse function of and vice-versa. the inverse of the function f(x) is written as f^-1read as 'f inverse ' and we have y=f(x) if and only if x = \(f^{-1}\) (y) for every x∈D(f) and every y∈R(f). let u consider a many-one function defined by the following arrow diagram.

Thus, the function h= {(1, 2), (2,2), (3, 3) }

Let, we interchange the domain set and range set of h. we will get a relation R which is given below.

R= {(2, 1), (2, 2), (3, 3) }. Then the arrow diagram of this relation is given in the adjoining figure.

In relation R, two ordered pairs (2,1) and (2, 2) have the same first element 2. So, R is not a function.

It is concluded that a function f: A→ B will have its inverse function \(f^{-1}\) B→ A if and only if f is a one -one onto function.

Definition: If f: A→ B is a one to one onto function from A to B, then there exist a function g: B→ A such that the range of f is the domain of g is called the inverse of 'f' denoted by \(f{-1}\) (f inverse) such that y= f(x) if and only x=\(f{-1}\) (y) for every x∈D (f) and every y∈R(f).

Simple Algebraic Functions

A function that can be defined as the root of a polynomial equation is known as an algebraic function. we shall discuss some example of functions defined on the set R real number onto itself and these functions f(x) are defined by means of an equation. for instance the algebraic functions. \begin{align}f: R→R\end{align}

$$f(x)= a_ox^n +a_1x^n-1+……………+ a_n-1 x+a_n$$

For all x€R where the right side is a polynomial of degree n.

We shall discuss some particular causes of this equation.

Constant Function: For n=0, we have f(x)=a_o. it is usually denoted by f(x)=c.

In other words, a function is said to be a constant function if all its function values are the same. The graph of constant function is a straight line parallel to the x-axis at a given distance

Linear functions: For n=1, we have $$f(x) = a_ox + a_1$$. It is usually denoted by y = ax+b

In another word, a function is said to be a linear function if the polynomial is degree one. The graph of it is a straight line with slope m=a and y-intercept = for instance, y=x+2 is an equation of a straight line.

Identity function: if a=1 and b=0 in the linear function f(x) = ax +b, then we have

$$F(x) = x$$ for all $$x€R$$

This function is called identity function and its graph is shown below. It bisects the angle between the axes of coordinates.

Quadratic function: for n = 2, we have $$f(x)= a_1x^2 + a_1x + a_2.$$ It is written as $$f(x)=ax^2 + bx + c$$

This is a quadratic function which is polynomial of degree 2 and its graph is a parabola.

For instance, $$f(x) = x^{2} + x -2, f(x) = 4x^2, f(x) = x^2 + 2$$, etc are quadratic functions.

The graph of these functions is shown below.

Cubic function : for n= 3, we have\((x) f(x) = a_ox^3 +a_1x^2 + a_2x + a_3\) It is written as \(f(x)=ax^3 + bx^2 +cx + d\)

This is a cubic function which is polynomial of degree 3. For instance $$f(x) = x^3$$ is a cubic function whose graph is shown alongside

Trigonometric functions

A function defined as the function which associates each angle with the definite real number.So, the domain of the trigonometric function is the set of angles and its co-domain is the set of real numbers. traditionally trigonometric functions are defined for angles of a triangle.But these trigonometric functions can be defined for angles of any magnitude.

The trigonometric ratios for angles such as 30⁰, 45⁰, 60⁰, etc. can be calculated with the help of elementary plane geometry. The following table shows the values of trigonometrical ratios from 0⁰ to 360⁰.

POLYNOMIALS

Polynomial is an algebraic expression consist of variables and coefficients, that involves only the operations of addition, subtraction, multiplication, and non-negative integer exponents. 2x, 5x² + 3, 2a³ - b² + 3x - 2, etc are the examples of polynomials.

Simple Operations on Polynomials

Multiplication of polynomials

In order to multiply two polynomials, following steps are used.

(i) Multiply each term of one polynomial by each term of the other and

(ii) Then add (or subtract) the like terms thus obtained.

(iii) Simplify and arrange the like terms in ascending order or descending order.

In multiplying two polynomials, the law of indices \(x^m×x^n=x^{m+n}\) is applied.

it is noted that the degree of the product of two polynomials is equal to the sum of the degrees of the polynomial factors.

Worked out with examples

Example 1.

If \(f(x)=3x^3v^4\: and\:g(x)=5x^2y^3\), find \(f(x).g(x)\)

Solution:

\begin{align*}f(x)\times g(x) &=3x^3y^4×5^2y^3=(3×5)\:(x^4×y^3) \\ &=15.x^{3+2}.y^{4+3}=15x^4y^7\end{align*}

Example 2.

If \(f(x)=x^2-3x+2\) and \(g(x)=x^3-x^2+2x+4\), find \(f(x).g(x)\).

Solution:

Horizantal method:

\begin{align*} f(x)\times g(x)&=(x^2-3x+2)(x^3-x^2+2x+4) \\ &=x^2(x^3-x^2+2x+4)-3x(x^3-x^2+2x+4)+2(x^3-x^2+2x+4) \\ &=x^5-x^4+2x^3+4x^2-3x^4+3x^3-6x^2-6x^2-12x+2x^3-2x^2+4x+8 \\ &=x^5+(-1-3)x^4+(2+3+1)x^3+(4-6-2)x^2 (-12+4)x+8 \\ &=x^5-4x^4+7x^3-4x^2-8x+8\end{align*}

Vertical Method:

\begin{align*}\frac{\frac{x^3-x^2+2x+4\\\times\:x^2-3x+2}{2x^3-2x^2+4x+8\\-3x^4+3x^3-6x^2-12x\\x^5-x^4+2x^3-4x^2\\}}{x^5-4x^4+7x^3-4x^2-8x+8}\\ \end{align*}

Division of Polynomials

We know from arithmetic how to divide ad integer by another smaller integer. If 30 is divided by 7, the quotient is 4 and the remainder is 2. i.e., \begin{align*}7)30(4 \\ \frac{-28}{2}\end{align*}

Here, we observe that \(30=7×4+2.\)

this relation ca be stated as follows.

Dividend= Divisor × Quotient + Reminder

Similarly, we can divide polynomials.

let \(f(x)\) and g(x) be two polynomials such that g(x) is a polynomial of smaller degree than that of \(f(x)\) and g(x)≠0. Then, there exist unique polynomials Q(x) and R(x) such that

\(f(x)=g(x).Q(x)+R(x)\)

where \(F(x)\)=dividend, g(x)=divisor, Q(x) is quotient and R(x) is remainder.

If R(x) = 0, then the divisor g(x) is a factor of the dividend f(x). The other factor of f(x) is the quotient Q(x).

The relation f(x) = g(x). Q(x) + R(x).

The following steps are used to divide a polynomial by the other:

(i) Arrange the divided f(x) and divisor g(x) in standard form i.e. generally descending powers of variable x.

(ii) Divided the first term divided f(x) by the first term of divisor g(x) to get the first term of quotient Q(x).

(iii) Multiply each term of divisor g(x) by the first term of quotient Q(x) obtained in step (ii) and subtract the product so obtained from the dividend f(x).

(iv) Take the remainder obtained in step (iii) as new dividend and continue the above process until the degree of the remainder is less than of the divisor.

Remainder Theorem

Statement: If a polynomial f(x) is divided by x - a, then the remainder is f(a).

Proof: If we divided f(x) by x - a, then we get Q(x) as quotient and R as remainder.

Then, f(x) = (x - a). Q(x) + R

Put x = a. Then,

or, f(a) = (a - a) Q (a) + R

or, R = f(a)

Hence, remainder = f(a) = the value of polynomial f(x) = a

Factor theorem

Statement: if f (x) is a polynomial and a is real number, then (x - a) is a factor of f(x) if f(a) = 0

Proof:If we divide f(x) by x-a, then we get Q(x) as quotient and R as remainder.

Then, f(x) = (x - a). Q(x) + R ...........(i)

Put x = a. Then

f(a) = (a - a). Q(a) + R

or, f(a) = R

When f(a) = 0. Then R = 0.

Putting the value of R in (i) we get,

f(x) = (x - a). Q(x)

So, (x - a) is a factor of f(x).

Hence, (x - a) is a factor of f(x) if f(a) = 0.

Synthetic Division

Synthetic division is the process which helps us to find the quotient and remainder when a polynomial f(x) is divided by x - a.

Application of synthetic division

Let Q(x) and R be the quotient and remainder when a polynomial f(x) is divided by ax -b.

Then, f(x) = (ax - b). Q(x) + R = a(x - \(\frac{b}{a}\)). Q(x) + R .

Where a.Q(x) = g(x)

or, Q(x) = \(\frac{1}{a}\) g(x).

Here, g(x) and R are the quotient and reminder when f(x) is divide by (x - \(\frac{b}{a}\)).

This result leads us to conclude that process of synthetic division discussed earlier is also useful to find out the quotient and remainder when f(x) is divided by (ax - b)

Factorization of a polynomial

Factor theorem and the synthetic division are very useful to find the factors of a polynomial.

Let us see the following example:

Factorize: x² + x - 2

Constant term of this polynomial is 4 and the possible factors of 4 are: ±1, ±2.

Since the degree of f(x) is 2, so there will be at most two factors.

When x = 1, f(1) = 1+1-2 =0

(x-1) is a factor.

When x = -1, f(1) = -1-1-2 =-4

(x+1) is not a factor.

when x = 2, f(2) = 4+2-2 = 4

(x+2) is not a factor

When x = -2, f(1) = 4-2-2 = 0

(x+2) is a factor.

∴ (x-1) and (x+2) is a factor.

∴ x² + x - 2 = (x-1)(x+2).

But instead of finding all the factors by using factor theorem, the synthetic division can be used after getting one with the help of factor theorem.

Polynomial Equation

Let f(x) = a_nxⁿ + a_n-1x^n-1 + ..... + a_o be the polynomial in x. Then f(x) = 0 is called a polynomial equation in x.

ax + b = 0 is a linear equation.

zx²+ bx + c = 0 is a quadratic equation.

ax\( ^3\) + bx² + cx + d = 0 is a cubic equation

ax\(^4\) + bx\(^3\) + cx² + dx + e = 0 is a big quadratic equation.

( Here, a,b,c,d,e are the real numbers).

If α is a real number such that f(α) = 0, thenα is called a root of the polynomial equation f(x) = 0.