# How do you find #f^-1(x)# given #f(x)=3/(x^2+2x)#?

The answer is

Then,

Comparing this equation to

Calculating the discriminant

So,

Therefore,

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To find the inverse function ( f^{-1}(x) ) given ( f(x) = \frac{3}{x^2 + 2x} ), follow these steps:

- Replace ( f(x) ) with ( y ): ( y = \frac{3}{x^2 + 2x} ).
- Swap the variables ( x ) and ( y ): ( x = \frac{3}{y^2 + 2y} ).
- Solve the equation for ( y ).
- Replace ( y ) with ( f^{-1}(x) ) to express the inverse function in terms of ( x ).

Following these steps:

- ( y = \frac{3}{x^2 + 2x} ).
- ( x = \frac{3}{y^2 + 2y} ).
- Solve for ( y ):
- ( x(y^2 + 2y) = 3 )
- ( xy^2 + 2xy - 3 = 0 )
- Solve this quadratic equation for ( y ).

- Once you find the expression for ( y ), replace ( y ) with ( f^{-1}(x) ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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