How to find the inverse function for a quadratic equation\[?\]

Hint:This question describes the operation of addition/ subtraction/ multiplication/ division. To solve this type of question we have to assume one equation in the form of a quadratic equation. Finally, we have to find the value of\[y\]from the quadratic equation. Also, we need to know the multiplication process with the involvement of square and square terms.

Complete step by step solution:
The given question is, we have to find the inverse function for a quadratic equation.

To solve the given question, we have to assume one equation in the form of a quadratic equation as follows, The basic form of a quadratic equation is

\[a{x^2} + bx + c = y\]\[ \to \left( 1 \right)\]

We assume, \[f\left( x \right) = {x^2} - 6x + 2\]

The term\[f(x)\]can be replaced by\[y\]. So, we get
\[y = {x^2} - 6x + 2 \to \left( 2 \right)\]

To solve the above equation we add and subtract\[{3^2}\]with the equation. So, the
equation\[\left( 2 \right)\]becomes,

\[y = {x^2} - 6x + 2 + {3^2} - {3^2}\]

The above equation can also be written as.
\[y = {x^2} - 6x + {3^2} + 2 - {3^2}\]
\[y = \left( {{x^2} - 6x + {3^2}} \right) + 2 - {3^2}\]\[ \to \left( 3 \right)\]

In the above equation, we have\[({x^2} - 6x + {3^2})\]. When it is compared to an algebraic formula we get,

\[\left( {{a^2} - 2ab + {b^2}} \right) = {\left( {a - b} \right)^2}\]
\[\left( {{x^2} - 2 \times x \times 3 + {3^2}} \right) = {\left( {x - 3} \right)^2}\]

So, we get
\[\left( {{x^2} - 6x + {3^2}} \right) = {\left( {x - 3} \right)^2}\]

Let’s substitute the above value in the equation\[\left( 3 \right)\], we get

\[y = {\left( {x - 3} \right)^2} + 2 - 9\]
\[y = {\left( {x - 3} \right)^2} - 7\]

For finding the inverse function of the above quadratic equation we have to
replace\[y\]with\[x\]and\[x\]with\[y\]. So, we get

\[y = {(x - 3)^2} - 7\]
\[ \downarrow \] \[ \downarrow \]
\[x = {(y - 3)^2} - 7\] (Inverse form)
Let’s solve the above equation,
\[x = {(y - 3)^2} - 7\]
It also can be written as
\[x + 7 = {(y - 3)^2}\]

Take square root on both sides of the above equation, we get
\[\left( {y - 3} \right) = \pm \sqrt {x + 7} \to \left( 4 \right)\]

Let’s find the\[y\]value from the above equation
\[y = 3 \pm \sqrt {x + 7} \to \left( 5 \right)\]

So, the final answer is, the inverse function of \[{x^2} - 6x + 2\]is\[y = 3 \pm \sqrt {x + 7} \]. By using the above-mentioned process we can find the inverse function of any quadratic equation.

Note: In this type of question if no equation is given we have to assume an equation in the basic form of a quadratic equation.

 To make easy calculation we would try to convert the equation in the form of algebraic formulae like\[{\left( {a - b} \right)^2}\],\[{\left( {a + b} \right)^2}\],\[\left( {{a^2} - {b^2}} \right)\], etc.

 To find the inverse function we have to replace the\[x\] term with\[y\]term and \[y\]term with\[x\].