Inverse Of Function Inverse Of Function In mathematics, an inverse function is a function that undoes another function: If an input x into the function ƒ produces an output y, then putting y into the inverse function g produces the output x, and vice versa. i.e., ƒ(x)=y, and g(y)=x. More directly, g(ƒ(x))=x, meaning g(x) composed with ƒ(x) leaves x unchanged. A function ƒ that has an inverse is called invertible; the inverse function is then uniquely determined by ƒ and is denoted by ƒ−1 (read f inverse, not to be confused with exponentiation). Definitions :- Instead of considering the inverses for individual inputs and outputs, one can think of the function as sending the whole set of inputs, the domain, to a set of outputs, the range. Let ƒ be a function whose domain is the set X, and whose range is the set Y. Then ƒ is invertible if there exists a function g with domain Y and range X, with the property: If ƒ is invertible, the function g is unique; in other words, there can be at most one function g satisfying this property. That function g is then called the inverse of ƒ, denoted by ƒ−1.
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Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which case the inverse relation is the inverse function. Not all functions have an inverse. For this rule to be applicable, each element y ∈ Y must correspond to no more than one x ∈ X; a function ƒ with this property is called one-to-one, or information-preserving, or an injection. In this if we inverse the function we will get the same result as q. We can find the inverse just by the swapping of value p and q and we get a new relation which represents the inverse of a function. But we have to note that it is not necessary that an inverse is always a function. Let us assume that we are given a function say f(p) and getting the output q then we will denote the inverse function with q input as f-1 (q). For understanding the inverse function in a better manner let us take another example. Assume a function f(p)=2p+6 then what will be its inverse function let’s take a look: f(p) = 2p + 6 p ----> 2*p --- > 2*p + 6 This function will be processed in this manner. For calculating its inverse just does the opposite of each task or inverse of each task like replace addition with subtraction and multiplication with division. f-1(q) will be (q - 6)/2
< -----
q-6
<---- q
So the inverse function f-1(q) of function f(p) is:
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f-1(q) = (q - 6)/2 Mathematics approach: step 1: step 2:
f(p) = 2p + 6 Put q at the place of f(p) q = 2p + 6
step 3: Take Integer value at left side q - 6 = 2p step 4: Divide left side by the coefficient of p (q - 6) /2 = p Step 5:
p = (q - 6) / 2
Step 6: Put f-1(q) at the place of p. f-1(q) = (q - 6) /2 This is the all about inverse of function.
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