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

Question

Vedantu Content Team · Accepted Answer

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, theequation$$\left( 2 \right)$$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 toreplace$$y$$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$$x$$.

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