Question

Given g(x) = $\frac{3}{x^2+2x}$ find g^-1(x)

Answer 1

We know that g(x) = \frac{3}{x^2+2x}

We have to find g^-1(x)  or inverse of g(x)

Inverse of g(x) can be determined by equating g(x) to y, and determining the value of x in terms of y

g(x) = y = \frac{3}{x^2+2x}

⇒ y × (x² + 2x) = 3

⇒ yx² + 2xy = 3

⇒ yx² + 2yx - 3 = 0

Determining the roots of x using:

x = $\frac{-b + \sqrt{b^{2} - 4ac}}{2a}$ OR x = $\frac{-b - \sqrt{b^{2} - 4ac}}{2a}$ , where a is coefficient of x², b is coefficient of x, and c is the constant

⇒ x = $\frac{-2y + \sqrt{4y^2 - 4(2y)(-3)}}{2y}$ OR x = $\frac{-2y - \sqrt{4y^2 - 4(2y)(-3)}}{2y}$

⇒ x = $\frac{-2y + \sqrt{4y^2 + 24y}}{2y}$ OR x = $\frac{-2y - \sqrt{4y^2 + 24y}}{2y}$

Hence, g^-1(x) = $\frac{-2y + \sqrt{4y^2 + 24y}}{2y}$ OR x = $\frac{-2y - \sqrt{4y^2 + 24y}}{2y}$

Answer 2

Answer:

We get:

$g^{-1}(x)=\dfrac{-x+\sqrt{x(x+3)}}{x}\ or\ g^{-1}(x)=\dfrac{-x-\sqrt{x(x+3)}}{x}$

Step-by-step explanation:

In order to find the inverse of a given function f(x) we follows following steps:

1)   Substitute f(x)=y

2)  Interchange x and y

3) Solve for y.

Here we have the function g(x) as follows:

$g(x)=\dfrac{3}{x^2+2x}$

Now, we substitute:

      $g(x)=y$

i.e.

$y=\dfrac{3}{x^2+2x}$

Now, we interchange x and y:

$x=\dfrac{3}{y^2+2y}\\\\x(y^2+2y)=3\\\\xy^2+2xy-3=0$

We know that the solution to the quadratic equation of the type:

$at^2+bt+c=0$

is given by:

$t=\dfrac{-b\pm \sqrt{b^2-4ac}}{2a}$

Here,

$a=x,\ b=2x\ and\ c=-3$

Hence, the solution is:

$y=\dfrac{-2x\pm \sqrt{(2x)^2-4\times x\times (-3)}}{2\times x}\\\\y=\dfrac{-2x\pm \sqrt{4x^2+12x}}{2x}\\\\y=\dfrac{-2x\pm 2\sqrt{x^2+3x}}{2x}\\\\y=\dfrac{-x\pm \sqrt{x^2+3x}}{x}$

i.e.

$y=\dfrac{-x+\sqrt{x(x+3)}}{x},\ y=\dfrac{-x-\sqrt{x(x+3)}}{x}$