It's kiran
Today short peiod because if T.A. so today we did one to one function.
Definition
Function is defined as a relation of x,y where for every x there is one and only one value of y assigend . A one to one funtion has another restriction added on to this : For every y which hass been assigned to the x's no other x has that y (for every y there is one and only one value x)
Examples -:
Question 1: Is function f defined by
a one to one function?
Solution to Question 1:
. Two different valuse in the damain namely %and 6 have same output, hence function f is not a one to one funcction.
Question 2: Is function g defined by
a one to one function?
Solution to Question 2 :
Consider any two different values in the domain of function g and check that their corresponding output are different. Hence function g is a one to one function
- Question 3: Is faction f given by
f(x) = -x 3 + 3 x 2 - 2 ,
a one to one function?
Solution to Question 3:
- A graph and the horizontal line test can help to answer the above question.
Since a horizontal line cuts the graph of f at 3 different points, that means that they are at least 3 different inputs x1, x2 and x3 with the same output Y and therefore f is not a one to one function. - Question 4: Show that all linear functions of the form
f(x) = a x + b ,
where a and b are real numbers such that a not equal to zero, are one to one functions.
Solution to Question 4:
- We start with f(A) = f(B) and show that this leads to a = b
a(A) + b = a(B) + b
- Add -b to both sides of the equation to obtain
a(A) = a(B)
- Divide both sides by a since it is not equal to zero to obtain
A = B
- Since we have proved that f(A) = f(B) leads to A = B then all linear functions of the form f(x) = a x + b are one-to-one functions.
A = B
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