Inverse Function :
Let f be a one-one onto function from A to B.
Let y be an arbitrary element of B.
Then f being onto, there exists an element x in A such that
f(x) = y
So, we may define a function, denoted as f-1 as
f-1 : B ---> A : f-1(y) = x
if and only if f(x) = y.
The above function f-1 is called the inverse of f.
A function is invertible, if and only if f is one-one onto.
More clearly, one-one onto function will have an inverse function
You may be able to find f-1 for any function f.
But, f-1 will also be a function, only if f is one-one onto function.
Remarks :
If f is one-one onto, then f-1 is also one-one onto.
Illustration :
If f : A ---> B, then
f-1 : B---> A
Let f(x) = x + k (k is a constant).
Step 1 :
In the above function, f(x) has to be replaced by y.
Then, we will get
y = x + k
y = x + k has been defined by "y" in terms of "x"
Step 2 :
Now we have to redefine y = x + k by "x" in terms of "y"
Then we will get
x = y - k
Step 3 :
In x = y - k, replace "x" by f-1 (x) and "y" by "x".
Therefore, inverse of f(x) is,
f-1(x) = x - k
Problem 1 :
Find the inverse of the function f(x) = 2x + 3
Solution :
Step 1 :
Given function : f(x) = 2x + 3
In the above function f(x) to be replaced by "y".
Then, we will get
y = 2x + 3
y = 2x + 3 has been defined by "y" in terms of "x"
Step 2 :
Now we have to redefine y = 2x + 3 by "x" in terms of "y".
y = 2x + 3
Subtract 3 from each side.
y - 3 = 2x
Divide each side by 2.
(y-3)/2 = x
x = (y-3) / 2
Now, the function has been defined by "x" in terms of "y".
Step 3 :
In x = (y - 3)/2, replace "x" by f-1 (x) and "y" by "x".
So, inverse of f(x) is,
f-1 (x) = (x - 3)/2
Problem 2 :
Find the inverse of the function h(x) = log10(x).
Solution :
Step 1 :
Given function : h(x) = log₁₀(x)
In the above function h(x) to be replaced by "y".
Then, we will get
y = log10(x)
y = log10(x) has been defined by "y" in terms of "x".
Step 2 :
Now we have to redefine y = log10(x) by "x" in terms of "y".
y = log10(x)
10y = x
x = 10y
Now, the function has been defined by "x" in terms of "y"
Step 3 :
In x = 10y replace "x" by h-1(x) and "y" by "x".
So, inverse of h(x) is,
h-1(x) = 10x
After having gone through the stuff given above, we hope that the students would have understood, how to find inverse of a function.
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