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Polynomial Function






Polynomial Function

Polynomial Function is defined as the real valued function $f : R \rightarrow R$ if for each $x \in R$ $y = f(x)=a_nx^n+a_{n-1}x^{n-1}+a_{n-2}x^{n-2}+.....+a_1x+a_0$
Where $a_n,a_{n-1},....,a_1,a_0$ are constant and real numbers and n is a non-negative integer
Some important points to remember
  • an,an-1,...,a,a0 are called the coefficients for xn,xn-1 ,..,x,x0
  • n is called the degree of the polynomial function
  • A constant Function is the polynomial function with zero degree
  • n is a non-negative integer.It can not be fraction also
Example of Polynomial functions
$3x^2 + 4x+6$
$9x^3 -4x^2 +1$
$4x +1$
$4x^2 + \sqrt {2} $

The below are not polynomial functions
$x^{-4} + 1$
$ \sqrt {x} + 2x +1$
$ x^{2/3} +1$

Types of Polynomial function based on degree

Type of Polynomial function
Example
Degree
Constant function
$1$
n=0
Linear function
$4x +1$
n=1
Quadratic function
$x^2 +4x +1$
n=2
Cubic Function
$3x^3 -4x^2+x +11$
n=3
Quartic Function
$x^4 +2x^3 +4x^2+x +1$
n=4
 
Function f(x) =0 is also an polynomial function with undefined degree

Domain and Range of the Polynomial Function



Domain = R
Range is dependent on the type of polynomial function.
For Linear function ,Range is R
For constant function Range is {c}
For Quadratic function like ${x^2 +1}$ , Range is $[1,\infty)$

Graph of the Polynomials Function

The graph of the Polynomial function depends on the type of polynomial functions.

For constant and Linear function

It is always straight line for constant and Linear Function
Here are the graph for function
$f(x)=3$ ( Constant function)
$f(x)=x+2$(Linear Function)
$f(x)=3x+1$(Linear Function)

graph of polynomial function(linear function)

For Quadratic function


For Quadratic function, the graph is a parabolic graph. Lets see few example

1) $f(x) =2x^2$ , $h(x) =5x^2$ , $g(x) =10x^2$
graph of polynomial function(upward quadratic function)
We can see the graph is a upward parabola and as we increase the coefficient of $x^2$,the graph is stretched vertically

2) $f(x) =-2x^2$ , $h(x) =-5x^2$ , $g(x) =-10x^2$
graph of polynomial function(downward quadratic function)
We can see the graph is a downward parabola and as we increase the coefficient of $x^2$,the graph is stretched vertically

3) $f(x) = x^2 + 2x +1$ , $g(x) = x^2 + 4x +1$ ,$h(x) =x^2 +6x +1$



We can see the graph is a upward parabola and with increase in coefficient of x ,it drifts downwards on the left

4. $f(x) =x^2 +1$ and $h(x) =-x^2 -1$

graph of polynomial function(quadratic function with no real roots)
Here since their are no real roots of the given quadratic function , the graph is not touching the x-axis. First graph is upward above x-axis and second graph is downward below x-axis
So , the intersection of parabola graph depends on the real roots of the function

5.$f(x) =x^2 -5x +6$ and $h(x) =-x^2 +7x -12$


Here since their are real roots of the given quadratic function ,the graph is intersecting the the x-axis.

For Cubic function



Lets see few example of the cubic function
1. $f(x) =x^3$
graph of polynomial function(cubic function)

We can see it extends in both upward and downward direction

2. $f(x) = x^3 + 1$
graph of polynomial function(cubic function with one real root)
Here is the cubic function with one real root

3.$f(x) = x^3 -3x^2 +3x -1= (x-1)^2$
graph of polynomial function(cubic function with three equal real root)
Here is the cubic function with three equal real root

4.$f(x) = x^3+12x^2+39x+28=(x+1)(x+4)(x+7)$
graph of polynomial function(cubic function with three different real root)
Here is the cubic function with three different real root.

For Quartic function


1. $f(x) =x^4 -5x^2 +4 =(x-1)(x-2)(x+1)(x+2)$
graph of polynomial function(Quartic function)
This is a quartic function and it has four real roots .

Some important points on Polynomial functions graphs

1. The turning point in a graph is defined as the points from where graph from upward to downward or downward to upward. The turning points in the graph is always less or equal to (n-1) of the polynomial function.So a quartic function has maximum 3 turning points in the graph.A quadratic equation has maximum one turning point. A Cubic equation has maximum 2 turns. You can validate these in above given example graph also

2. The zero's or root of the polynomial function are point at which graph intersect x-axis , i,e the point where the value of y=0. The real roots of the polynomial function is always less or equal to the degree n of the polynomial.


Solved examples of Polynomial Functions

1. Find the domain of the below function?
a. $y =x^2 +5$
b. $y = x^3 $
c. $y=x^2 -7x +5$ Solution

a. Clearly it is defined for all $x \in R$ b. Clearly it is defined for all $x \in R$ c. Clearly it is defined for all $x \in R$ 2. Find the domain and range of function
$y= \frac {1}{\sqrt {x- [x]}}$
Solution
We have ,
$y= \frac {1}{\sqrt {x- [x]}}$
Domain of y
We know that
$0 \leq x -[x] < 1 , x \in R$
for $x \in Z ,x -[x]=0$
So Domain is R - Z
Range of y
$0 < x- [x] < 1 , x \in R -Z$
$ 0 < \sqrt {x -[x]} < 1$
$ 1 < \frac {1}{\sqrt {x- [x]}} < \infty$
So Range is = $(1,\infty)$

3. Let $f : R \rightarrow R$ , $f(x) = x^2 + 2[x] -1$ for each $x \in R$
Find the values of f(x) at x= 1.2 ,-.5 ,-2.1
Solution
$f(x) = x^2 + 2[x] -1$
$f(1.2) = (1.2)^2 + 2[1.2] -1 =1.44 +2-1=2.44$
$f(-.5) = (-.5)^2 + 2[-.5] -1 =.25 -2-1=-2.75$
$f(-2.1) = (-2.1)^2 + 2[-2.1] -1 =4.41 -6-1=-2.59$

Quiz Time

Question 1Find the domain of the function $f(x)= \frac {1}{\sqrt {[x]-x}}$
A. R - Z
B. R
C. {}
D. Z
Question 2Find the range of values of x for which $[x]^2 -4 =0$ $
A. $[-2,-1) \cup [2,3)$
B. $[-2,-1] \cup [2,3]$
C. $[-2,-3) \cup [2,3)$
D. $[-2,-3] \cup [2,3)$
Question 3 if $f(x) =[x]^2 +[x] +2$,then which of them is incorrect
A. f(1)=4
B. f(.75)=2
C. f(-.50)=2
D. f(0) =3
Question 4 Value of the expression [{x}] + {[y]} is
A. 1
B. 0
C. 2
D. Cannot be determined
Question 5 The value {-.75} where {.} is the fractional part is
A. -.75
B. .25
C. 0
D. None of these
Question 6The domain of the function f given by $f (x) = \frac {1}{\sqrt {x^2 -[x]^2}}$ is
A. Domain = R
B. Domain = R+
C. Domain = R+ -Z
D. Domain = Z



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