Solve f_{n}1/f_{1}(x)=1/1-f_{n(x)}

$$f _ { n } \frac { 1 } { f _ { 1 } } ( x ) = \frac { 1 } { 1 - f _ { n } ( x ) }$$

Solve for f_1

$$f_{1}=f_{n}x\left(1-f_{n}x\right)$$

$$x\neq 0\text{ and }f_{n}\neq \frac{1}{x}\text{ and }f_{n}\neq 0$$

Steps for Solving Linear Equation

Variable f_{1} cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f_{1}\left(f_{n}x-1\right), the least common multiple of f_{1},1-f_{n}x.

$$f_{n}\left(f_{n}x-1\right)\times 1x=-f_{1}$$

Use the distributive property to multiply f_{n} by f_{n}x-1.

$$\left(xf_{n}^{2}-f_{n}\right)\times 1x=-f_{1}$$

Use the distributive property to multiply xf_{n}^{2}-f_{n} by 1.

$$\left(xf_{n}^{2}-f_{n}\right)x=-f_{1}$$

Use the distributive property to multiply xf_{n}^{2}-f_{n} by x.

$$f_{n}^{2}x^{2}-f_{n}x=-f_{1}$$

Swap sides so that all variable terms are on the left hand side.

$$-f_{1}=f_{n}^{2}x^{2}-f_{n}x$$

Divide both sides by -1.

$$\frac{-f_{1}}{-1}=\frac{f_{n}x\left(f_{n}x-1\right)}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$f_{1}=\frac{f_{n}x\left(f_{n}x-1\right)}{-1}$$

Divide f_{n}x\left(f_{n}x-1\right) by -1.

$$f_{1}=-f_{n}x\left(f_{n}x-1\right)$$

Variable f_{1} cannot be equal to 0.

$$f_{1}=-f_{n}x\left(f_{n}x-1\right)\text{, }f_{1}\neq 0$$

Solve for f_n (complex solution)

$$f_{n}=-\frac{\sqrt{1-4f_{1}}-1}{2x}$$

$$f_{n}=\frac{\sqrt{1-4f_{1}}+1}{2x}\text{, }x\neq 0\text{ and }f_{1}\neq 0$$

Solve for f_n

$$f_{n}=-\frac{\sqrt{1-4f_{1}}-1}{2x}$$

$$f_{n}=\frac{\sqrt{1-4f_{1}}+1}{2x}\text{, }x\neq 0\text{ and }f_{1}\leq \frac{1}{4}\text{ and }f_{1}\neq 0$$