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### High School Mathematics - 24.7 Inverse of Functions

 Consider the function of of the set of children to the set of their mothers. It is obviously a many-one type function as several children may have the same mother. Its inverse however will not exist. In this case we will have a relation in which several ordered pairs of the form (mother, child) may have the same first component. Consider the functions f1, f2, f3, f4, f5 exhibited by the following diagrams:     f1(one-one into type), f1-1 doesn't exist. f2(one-one onto type), f2-1 doesn't exist. f3(many-one into type), f3-1 doesn't exist. f4(many-one onto type), f4-1 doesn't exist. f5(constant type), f5-1 doesn't exist. The above illustrations lead to a conclusion that A function f: A -> B will have its inverse g: B -> A if and only if f is a one-one function. f is an onto function i.e f is bijective. Definition: If f is a function f: A - >B, then there will exist a function g: B ->A if f is a one-one onto function, and such that the range of f is the domain of g and domain of f is the range of g; then g is called the inverse of f and is denoted by f-1. Also f is called the inverse of g and is denoted as g-1. Symbolically, a function f: x - >y , then its inverse is represented as f-1: y - > x, or if y = f(x), then its inverse is represented as f-1(y) = x. Example: If f: R -> R be define by f(x) = x3 + 7, find a formula that defines f-1. Solution: Method to find the inverse of a bijective function Step 1: Put f(x) = y, where y � B and x �A. Step 2: Solve f(x) = y to obtainx in terms of y. Step 3: In the relation obtained in step 2, replace x by f-1 to obtain the inverse of f. Step 4: Let f(x) = x3 + 7 = y, then x3 = y - 7 Step 5: x = (y-7)1/3 = >f-1(y) = (y - 7)1/3 Answer: Hence f-1 : R -> R : f-1(x) = (x-7)1/3for all x � R. Directions: Solve the following problems. Also write at least 5 examples of your own.
 Q 1: If f: R ->R is such that f(x) = log3x, f-1 is equal to 31/xlog x33x Q 2: If f(x) = (2x+1)/3, for each x � R, find f-1.(-2x-1)/3(3x-1)/2(2x-1)/3 Q 3: Let f : A -> B, find f-1(B).Inverse doesn't existAB Q 4: If f : R+ -> R such that f(x) = log8x,find f-1(x).8y8xlog8y Q 5: If f: R ->R be defined by f(x) = 3x- 4, then f-1(x) is 3x + 4(x+4)/3x/3 - 4 Q 6: If f: R=>R such that f(x) = e3x+2,find f-1e2501 Q 7: If f:R -> R, write the inverse of f(x) = 3x + y.(y-4)/3(x-4)/3-(3x+y) Q 8: f: R -> R be defined by f(x) = 10x - 7. if g = f-1 then g(x) is 1/(10x - 7)1/(10x + 7)(x + 7)/10 Question 9: This question is available to subscribers only! Question 10: This question is available to subscribers only!

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