**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) = log_{10}(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 = log_{10}(x)

y = log_{10}(x) has been defined by "y" in terms of "x".

**Step 2 :**

Now we have to redefine y = log_{10}(x) by "x" in terms of "y".

y = log_{10}(x)

10^{y} = x

x = 10^{y}

Now, the function has been defined by "x" in terms of "y"

**Step 3 :**

In x = 10^{y} replace "x" by h^{-1}(x) and "y" by "x".

So, inverse of h(x) is,

h^{-1}(x) = 10^{x}

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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