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4. Into function
There is at least one element of B which has no pre-image. In above fig.(i) the function is one-one and into, while in fig.(ii) the function is many-one and into. For types of functions, the four arrow diagrams given for one-one and many-one are repeated for ONTO and INTO functions because e..
There is at least one element of B which has no pre-image. In above fig.(i) the function is one-one and into, while in fig.(ii) the function is many-one and into. For types of functions, the four arrow diagrams given for one-one and many-one are repeated for ONTO and INTO functions because e..Continuity at a Point
A function f (x) is said to be continuous at x = a ifA function f (x) is said to be continuous at x = a if f (a) exist..
A function f (x) is said to be continuous at x = a ifA function f (x) is said to be continuous at x = a if f (a) exist..Note 3:
Every polynomial function is continuou..
Theorem 10:
If f and g are real functions such that fog is defined, if g is continuous at a point c, and if f is continuous at g(c), then fog is continuous at ..
Step I:
Factor of f(x) and g(x..
Proof:
Consider a circle with centre O and radius r. Join AB. Let the tangent at B meet OA produced at P. Draw BN perpendicular to OA. From ONB, BN = r sin q From OBP, BP = r tan q From the figure, we have Area of triangle OAB < Area of sector OAB < Area of triangle OBP ..
Consider a circle with centre O and radius r. Join AB. Let the tangent at B meet OA produced at P. Draw BN perpendicular to OA. From ONB, BN = r sin q From OBP, BP = r tan q From the figure, we have Area of triangle OAB < Area of sector OAB < Area of triangle OBP ..Proof:
We know that, Further, we have Substituting this value in (2), we have From (1) and (3), we have ..
We know that, Further, we have Substituting this value in (2), we have From (1) and (3), we have ..Proof:
..
..Infinite limits
Let f(x) be a function of x, if the value of f(x) can be made greater than any pre-assigned number by taking x close to 'a', then we say Similarly, if the value of f (x) can be made less than any pre-assigned number by taking x close to 'a', then we say f (x) tends to - as x approaches 'a'..
Let f(x) be a function of x, if the value of f(x) can be made greater than any pre-assigned number by taking x close to 'a', then we say Similarly, if the value of f (x) can be made less than any pre-assigned number by taking x close to 'a', then we say f (x) tends to - as x approaches 'a'..See what our Users say :
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