Question

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

Answer 1

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$$x$$.