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Ainat [17]
3 years ago
9

Given g(x) =

rac{3}{x^2+2x}" align="absmiddle" class="latex-formula"> find g^-1(x)
Mathematics
2 answers:
sweet [91]3 years ago
7 0

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}

liraira [26]3 years ago
6 0
<h2>Answer:</h2>

We get:

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

<h2>Step-by-step explanation:</h2>

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}

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