Question

find the range of the function fx=
$x {}^{2} + 2x + 1 \div x {}^{2} - 8x + 12$
​

Answer 1

$\large\underline{\bold{Given \:Question - }}$

$\sf \: Find \: the \: range \: of \: the \: function \: f(x) = \dfrac{ {x}^{2} + 2x + 1}{ {x}^{2} - 8x + 12 }$

$\large\underline{\bold{Solution-}}$

Following steps for used finding the range of a function are:

Write down y = f(x) and then solve the equation for x, giving something of the form x = g(y).

Find the domain of g(y), and this will be the range of f(x).

Note :-

If you can't seem to solve for x, then try graphing the function to find the range.

Now, Let's solve the problem now!!

Step :- 1

$\sf \: Let \: f(x) = y \: = \: \dfrac{ {x}^{2} + 2x + 1}{ {x}^{2} - 8x + 12 }$

$\sf \: {yx}^{2} - 8xy + 12y = {x}^{2} + 2x + 1$

$\sf \: {yx}^{2} - 8xy + 12y - {x}^{2} - 2x - 1 = 0$

$\sf \: (y - 1) {x}^{2} - x(8y + 2) + 12y - 1 = 0$

Step :- 2

Discriminant of the above quadratic expression is

$\sf \: Discriminant, D = {b}^{2} - 4ac$

$\sf \: D = {(8y + 2)}^{2} - 4(y - 1)(12y - 1)$

$\sf \: = 4 {(4y + 1)}^{2} - 4( {12y}^{2} - y - 12y + 1)$

$\sf \: = 4\bigg( {16y}^{2} + 1 + 8y - 12 {y}^{2} + 13y - 1 \bigg)$

$\sf \: =4( {4y}^{2} + 21y)$

Now, solution of above quadratic equation is given by

$\rm :\longmapsto\:\sf \: x \: = \: \dfrac{ - b \: \pm \: \sqrt{ {b}^{2} - 4ac} }{2a}$

$\rm :\longmapsto\:x = \dfrac{8y + 2 \: \pm \: \sqrt{4y(4y + 21)} }{2(y - 1)}$

$\rm :\longmapsto\:x = \dfrac{4y + 1 \: \pm \: \sqrt{y(4y + 21)} }{(y - 1)}$

$\rm :\longmapsto\:y(4y + 21) \geqslant 0$

$\bf\implies \:y \leqslant - \dfrac{21}{4} \: \: \: or \: \: \: y \geqslant 0$