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

f^-1(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(x)^-1 = √x

Example 1 :

Find the inverse of the quadratic function and graph it.  

f(x) = x²

Solution :

Replace f(x) by y.

y = x²

Interchange x and y. 

x = y²

y² = x

Solve for y.

Take square root on both sides. 

y = ±√x

Replace y by f^-1(x).

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.

Example 2 :

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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