Tuesday, July 10, 2012
Monday, July 9, 2012
Rational Functions
A rational function is a function that is a fraction of the form f(x) = P(x)/Q(x) , where p(x) and q(x) are polynomials and q(x) does not equal zero e.g. f(x) = 1/(x+2) , g(x) = 1/(x2-x-2) are examples of rational functions where f and g are the names of the functions and x is the independent variable.
The domain of a rational function consists of all real numbers x except those for which the denominator is zero. When graphing a rational function, we must pay special attention to the behavior of the graph near those x-values. Let us find domain of f(x) = 1/(x+3)
x = -3
Enter different rational functions in the input box of the applet below and try to find the domain , range and asymptote.
The domain of a rational function consists of all real numbers x except those for which the denominator is zero. When graphing a rational function, we must pay special attention to the behavior of the graph near those x-values. Let us find domain of f(x) = 1/(x+3)
- The function has independent variable (x) in the denominator
- Set the denominator equal to zero
x = -3
- The domain of function f is all real numbers except x = -3
Enter different rational functions in the input box of the applet below and try to find the domain , range and asymptote.
Sunday, July 8, 2012
Polynomials - Class IX
POLYNOMIALS
Practice Questions – Class IX
- Find the remainder when x3-ax2+6x-a is divided by x-a.
- Find the value of a and b so that the polynomial x3+10x2+ax+b is exactly divisible by (x-1) as well as by (x+2)
- Find the value of ‘a’ so that x+6 is a factor of x3+3x2+4x+a
- Show that (x+a) is a factor of xn+an for any odd positive integer n.
- Find the value of p if (x-p) is a factor of the polynomial x6-px5+x4-px3+3x-p+2.
- Show that 3 is a zero of the polynomial x3-8x2+8x+21.
- Prove that x2+2x+3 has no zero.
- Find the integral zeroes of the polynomial 2x3+5x2-5x-1.
- Find the values of a and b so that the polynomial x3-ax2-13x+b has (x-1) and (x+3) as factors.
- If x3+mx2+nx+6 has (x-2) as a factor and leaves remainder 3 , when divided by (x-3) , find the values of m and n.
- Find the value of p and q so that x-1 and x+1 are factors of x4+px3+2x2-3x+q.
- Find a & b such that x+2 and x-2 are factors of the polynomials ax4+2x3-3x2+bx-4
- If x-3 and x-1/3 are both factors of ax2+5x+b, show that a=b
- Find the value of a3-8b3-36ab-216 when a = 2b+6
- The polynomials ax3+3x2-3 and 2x3-5x+a when divided by x-4 leaves the remainders p and q respectively. Find the value of a if 2p=q.
- Prove that (x+y) 3+(y+z) 3+(z+x) 3-3(x+y)(y+z)(z+x)=2(x3+y3+z3-3xyz)
- Show that x-y , y-z and z-x are factors of x2 (y-z)+y2 (z-x)+z2 (x-y)
- Find the value of p and q so that x-1 and x+1 are factors of x4+px3+2x2-3x+q.
- Find a & b such that x+2 and x-2 are factors of the polynomials ax4+2x3-3x2+bx-4
- If x-3 and x-1/3 are both factors of ax2+5x+b, show that a=b
- The polynomials ax3+3x2-3 and 2x3-5x+a when divided by x-4 leaves the remainders p and q respectively. Find the value of a if 2p=q.
- Let A and B be the remainders when polynomials x3+2x2-5ax-7 and x3+ax2-12x+6 are divided by x+1 and x-2 respectively. If 2A+B=6 , find the value of a.
- Without actually calculating the cubes , evaluate each of the following
Saturday, July 7, 2012
Absolute Value Function
The absolute value (or modulus) | a | of a real number a is the numerical value of a without regard to its sign. So,for example,the absolute value of 2 is 2,and the absolute value of –2 is also 2. The absolute value of a number may be thought of as its distance from zero.
From the above definition we can say that the absolute value of a is always either positive or zero, but never negative. If x is greater than or equal to zero, then use |x| = x , i.e., if a number is non negative, then its absolute value is itself. Whereas , If x is less than zero, then use |x| = - x i.e. if a number is negative, then its absolute value is its opposite.
The function f(x) = |x| is called the absolute value function. The domain of the absolute value function is the set of real numbers and the range is the set of positive real numbers.
In GeoGebra , absolute value function is written as abs(function name). You can enter the following functions in the input box of the applet and see the nature of graph.
1. f(x) = 3 abs(x-2) + 1 2.f(x) = -2 abs(x+2) + 3 3.f(x) = abs(x-1) 4.f(x) = 0.5 abs(x+1) - 2 5.f(x) = 3 - 2 abs(x-1)
Try to find the domain and range in each of the above cases with the movement of points B and C.
For any real number a the absolute
value or modulus of a is denoted
by | a | and is defined as
From the above definition we can say that the absolute value of a is always either positive or zero, but never negative. If x is greater than or equal to zero, then use |x| = x , i.e., if a number is non negative, then its absolute value is itself. Whereas , If x is less than zero, then use |x| = - x i.e. if a number is negative, then its absolute value is its opposite.
The function f(x) = |x| is called the absolute value function. The domain of the absolute value function is the set of real numbers and the range is the set of positive real numbers.
In GeoGebra , absolute value function is written as abs(function name). You can enter the following functions in the input box of the applet and see the nature of graph.
1. f(x) = 3 abs(x-2) + 1 2.f(x) = -2 abs(x+2) + 3 3.f(x) = abs(x-1) 4.f(x) = 0.5 abs(x+1) - 2 5.f(x) = 3 - 2 abs(x-1)
Try to find the domain and range in each of the above cases with the movement of points B and C.
Friday, July 6, 2012
Domain and Range of a Function
In mathematics, a function is a relation between a set of inputs and a set of potential outputs with the property that each input is related to exactly one output. An example of such a relation is defined by the rule f(x) = x2, which relates an input x to its square, which are both real numbers. The output of the function f corresponding to an input x is denoted by f(x) (read "f of x"). If the input is –3, then the output is 9, and we may write f(–3) = 9.
Thus we have two quantities (called "variables") and we observe that there is a relationship between them. If we find that for every value of the first variable there is only one value of the second variable, then we say: "The second variable is a function of the first variable."
The first variable is the independent variable (usually written as x), and the second variable is the dependent variable (usually written as y).
The domain of a function is the set of all possible x-values which will make the function work and will output real y-values.
The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.
The following applet shows the graph of sine function. Its domain ( which can be tracked by point B) and range(which can be tracked by point C) are respectively (-∞ , ∞) and [-1,1].
Thus we have two quantities (called "variables") and we observe that there is a relationship between them. If we find that for every value of the first variable there is only one value of the second variable, then we say: "The second variable is a function of the first variable."
The first variable is the independent variable (usually written as x), and the second variable is the dependent variable (usually written as y).
The domain of a function is the set of all possible x-values which will make the function work and will output real y-values.
The range of a function is the possible y values of a function that result when we substitute all the possible x-values into the function.
The following applet shows the graph of sine function. Its domain ( which can be tracked by point B) and range(which can be tracked by point C) are respectively (-∞ , ∞) and [-1,1].
Subscribe to:
Posts (Atom)