# INVERSE OF A QUADRATIC FUNCTION

Inverse of a quadratic function :

The general form of a quadratic function is

f(x) = ax² + bx + c

Then, the inverse of the above quadratic function is

f¹(x)

For example, let us consider the quadratic function

g(x) = x²

Then, the inverse of the quadratic function is g(x) = x² is

g⁻¹(x)  =  √x

## Finding inverse of a quadratic function

We have to apply the following steps to find inverse of a quadratic function

Step 1 :

Let f(x) be a quadratic function

In the above function,

f(x) to be replaced by "y" or y = f(x)

So, y  =  quadratic function in terms of "x"

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

Step 2 :

Now, we have to redefine the function y = f(x) by "x" in terms of "y"

Then we will get x = g(y)

Step 3 :

In x = g(y), replace "x" by f⁻¹(x) and "y" by "x".

Hence inverse of f(x) is,  f⁻¹(x) = g(x)

## Finding inverse of a quadratic function - Examples

Example 1 :

Find the inverse and its graph of the quadratic function given below.

f(x)  =  x²

Solution :

Step 1 :

In the given function, let us replace f(x) by "y". Then, we have

y  =  x²

Step 2 :

We have to redefine y = x² by "x" in terms of "y". Then we have

√y  =  x   or   x  =  √y

Step 3 :

In x = y, replace "x" by f⁻¹(x) and "y" by "x".

Hence inverse of f(x) is

f⁻¹(x) = √x

Graphing the inverse of f(x) :

We can graph the original function by plotting the vertex (0, 0). The parabola opens up, because "a" is positive.

And we get f(1)  =  1 and f(2)  =  4, which are also the same values of f(-1) and f(-2) 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, (2, 4) becomes (4, 2).

We have to do this because the input value becomes the output value in the inverse, and vice versa.

The graph of the inverse is a reflection of the original function about the line y = x.

Graph of f(x) and its inverse f⁻¹(x)

Example 1 :

Find the inverse and its graph of the quadratic function given below.

f(x)  =  2(x + 3)² - 4

Solution :

Step 1 :

In the given function, let us replace f(x) by "y". `Then, we have

y  =  2(x + 3)² - 4

Step 2 :

We have to redefine y = x² by "x" in terms of "y". Then we have

Step 3 :

Replacing "x" by f⁻¹(x) and "y" by "x" in the last step, we get inverse of f(x)

Hence inverse of f(x) is

Graphing the inverse of f(x) :

We can graph the original function by taking (-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.

Graph of f(x) and its inverse f⁻¹(x)

After having gone through the stuff given above, we hope that the students would have understood "Inverse of a quadratic function".

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