# HOW TO FIND INVERSE OF A FUNCTION

## About the topic "How to find inverse of a function"

"How to find inverse of a function?" is the question having had by almost all the students who study math in high schools.

Even though students can get this stuff on internet, they do not understand exactly what has been explained.

To make the students to understand "How to find inverse of any function", we have given step by step explanation.

## Steps involved in "How to find inverse of a function"

**Step 1 :**

Let **f(x) = x + k** ("k" is a constant).

In the above function **f(x)** 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"**.

Hence inverse of f(x) is, **f ⁻¹ (x) = x - k **

## Examples

To have better understanding of the steps explained above, let us look at some examples.

**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 ===> y - 3 = 2x

===> (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"**.

Hence inverse of f(x) is, **f ⁻¹ (x) = (x - 3)/2**

**Problem 2 : **

Find the inverse of the function **h(x) = log****₁****₀(x)**

**Solution :**

**Step 1 :**

Given function : **h(x) = log****₁****₀(x)**

In the above function **h(x)** to be replaced by **"y"**.

Then, we will get **y = ****log****₁****₀(x)** .

**y = ****log****₁****₀(x)** has been defined by **"y"** in terms of **"x" **

**Step 2 :**

Now we have to redefine **y = ****log****₁****₀(x)** by **"x"** in terms of **"y" **

y = log_{10}(x) ===> 10^{y} = x
or **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"**

Hence inverse of h(x) is, **h **^{-1}(x) = 10^{x}