Chapter 0.5, Problem 9E

To determine

Whether the function has inverse.

Answer to Problem 9E

The inverse of the function of the function $y=x^2-4x+6$ does not exist.

Explanation of Solution

Given data:

The function is $y=x^2-4x+6$ or $f\left(x\right)=x^2-4x+6$ .

Method/Formula used:

Substitute y for $f\left(x\right)$ and find x in terms of y , write x as $f^{-1}\left(y\right)$ , then replace y by x throughout.

Calculation:

The given function is $f\left(x\right)=x^2-4x+6$ .

Substitute y for $f\left(x\right)$ and solve for x as:

  $y=x^2-4x+6$

That further can be written as a quadratic equation $x^2-4x+6-y=0$ which can be solved for x using quadratic formula, for $a{x}^2+bx+c=0$ that is

  $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$ … (1)

Here, $a=1,b=-4,c=6-y$ , therefore, substitute values of a , b , and c in the formula (1)

  $\begin{array}{l}x=\frac{-\left(-4\right)\pm \sqrt{{\left(-4\right)}^2-4\left(1\right)\left(6-y\right)}}{2\left(1\right)}\\ x=\frac{4\pm \sqrt{16-24+4y}}{2}\\ x=\frac{4\pm 2\sqrt{y-2}}{2}\\ x=2\pm \sqrt{y-2}\end{array}$

Replace, x by $f^{-1}\left(y\right)$ ,

  $f^{-1}\left(y\right)=2\pm \sqrt{y-2}$

Change variable − replace y by customary variable x or another,

  $f^{-1}\left(x\right)=2\pm \sqrt{x-2}$

Thus, there are two inverses of function $f\left(x\right)$ given by $f^{-1}\left(x\right)=2-\sqrt{x-2}$ , and $f^{-1}\left(x\right)=2+\sqrt{x-2}$ . But inverse of a function is unique.

Hence, the inverse of the function of the function $y=x^2-4x+6$ does not exist.