Question

How to find the inverse function for a quadratic equation$?$

Answer 1

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$x$.