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
How to find inverse of a function f(x) :
Step 1 :
Replace f(x) by y.
Step 2 :
Interchange the variables x and y.
Step 3 :
Solve for y.
Step 4 :
Replace y by f^{-1}(x).
Example 1 :
Find the inverse of the function f(x) = 2x + 3.
Solution :
f(x) = 2x + 3
Replace f(x) by y.
y = 2x + 3
Interchange x and y.
x = 2y + 3
Solve for y.
x - 3 = 2y
y = (x - 3)/2
Replace y by f^{-1}(x).
f^{-1}(x) = (x - 3)/2
f^{-1} (x) = (x - 3)/2
Example 2 :
Find the inverse of the function f(x) = x^{2}.
Solution :
Replace f(x) by y.
y = x^{2}
Interchange x and y.
x = y^{2}
y^{2} = x
Solve for y. Taking square root on both sides,
y = ±√x
Replace y by f^{-1}(x).
f^{-1}(x) = ±√x
Example 3 :
Find the inverse of the function h(x) = log_{10}(x).
Solution :
h(x) = log_{10}(x)
Replace h(x) by y.
y = log_{10}(x)
Interchange x and y.
x = log_{10}(y)
Solve for y.
y = 10^{x}
Replace y by h^{-1}(x).
h^{-1}(x) = 10^{x}
Example 4 :
Find the inverse of the quadratic function and graph it.
f(x) = 2(x + 3)^{2} - 4
Solution :
Replace f(x) by y.
y = 2(x + 3)^{2} - 4
Interchange x and y.
x = 2(y + 3)^{2} - 4
Solve for y.
x + 4 = 2(y + 3)^{2}
(x + 4)/2 = (y + 3)^{2}
Take square root on both sides.
±√[(x + 4)/2] = y + 3
±√[(x + 4)/2] - 3 = y
y = -3 ± √[(x + 4)/2]
Replace y by f^{-1}(x).
f^{-1}(x) = -3 ± √[(x + 4)/2]
Graphing the inverse of f(x) :
We can graph the original function by plotting the vertex (-3, -4). The parabola opens up, because a is positive.
And we get f(-2) = -2 and f(-1) = 4, which are also the same values of f(-4) and f(-5) respectively.
To graph f^{-1}(x), we have to take the coordinates of each point on the original graph and switch the x and y coordinates.
For example, (-1, 4) becomes (4, -1).
We have to do this because the input value becomes the output value in the inverse, and vice versa.
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