INVERSE FUNCTIONS: LET f(x)=ln(x+3). 1a)SKETCH THE MIRROR LINE OF Y=X ON AN AXES

Question

INVERSE FUNCTIONS: LET f(x)=ln(x+3). 1a)SKETCH THE MIRROR LINE OF Y=X ON AN AXES and SKETCH AND LABEL THE GRAPH OF f(x). 1b) INDICATE THE DOMAIN AND RANGE OF f(x) and IS THE FUNCTION A 0NE-TO-ONE FUNCTION AND EXPLAIN GRAPHICALLY? 1c)FIND THE INVERSE FUNCTION OF F(X) AND SKETCH AND LABEL THE INVERSE FUNCTION ON THE AXES=COMPARE THE DOMAIN AND RANGE OF f(x) to the inverse function f(x)'s doman and range?

Explanation / Answer

Injective Functions: It is also called as one to one function. It satisfies the property that if f(a) = f(b) then, a = b. This is the condition for injective function Surjective Functions: Surjective function is also called as onto function. Here, every elements in the range is associated with atleast one element in the domain. If y is element in range then there is a x such that y = f(x). Bijective Functions: If the given function is both injective and surjective or one to one and onto then, it is called as a bijective function. Inverse Functions: If f is a function from A to B then inverse of a function is the function defined from B to A.It is denoted by f -1. That is, if f:A -> B then f -1 : B -> A. If f(x) = y then x = f-1(y). Solved Examples Question 1: Check whether the given function is bijective or not f(x)= x + 2 Solution: f(x) = x + 2 f(a) = a + 2 f(b) = b + 2 If f(a) = f(b), a + 2 = b + 2 a = b Therefore, it is one to one y = f(x) y is directly dependent on x. So, for every value of y there is a x value. Hence, it is onto. So, given function is Bijective. Question 2: Find the inverse of the function f(x) = 2x + 1 Solution: f(x) = 2x + 1 y = 2x + 1 2x = y - 1 x = (y - 1)/2 Inverse function is f-1(x) = (x - 1)/2 Question 3: Is the given function surjective f(x) = 2x + 1 where, f is function from R to R Solution: y = f(x) y = 2x + 1 For all real values of x, we get a real value of y. So, for every y, there is a x associated with it. So, the given function is onto or surjective. Question 4: f(x) = x2 + 2.Show that it is one-one. Solution: f(x) = x2 + 2 f(a) = a2 + 2 f(b) = b2 + 2 f(a) = f(b) a2 + 2 = b2 + 2 a2 = b2 (then take square root on both side) a = b Hence, it is one to one or injective. 2x = y - 1 x = (y - 1)/2 Inverse function is f-1(x) = (x - 1)/2

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