Solve f{1-x/x}=1/x+x^2/2x^2-2x+1-1

$$f \{ \frac { 1 - x } { x } \} = \frac { 1 } { x } + \frac { x ^ { 2 } } { 2 x ^ { 2 } - 2 x + 1 } - 1$$

Solve for f

$$f=-\frac{x^{3}-4x^{2}+3x-1}{\left(x-1\right)\left(-2x^{2}+2x-1\right)}$$

$$x\neq 1\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x\left(2x^{2}-2x+1\right), the least common multiple of x,2x^{2}-2x+1.

$$f\left(2x^{2}-2x+1\right)\left(1-x\right)=2x^{2}-2x+1+xx^{2}+x\left(2x^{2}-2x+1\right)\left(-1\right)$$

To multiply powers of the same base, add their exponents. Add 1 and 2 to get 3.

$$f\left(2x^{2}-2x+1\right)\left(1-x\right)=2x^{2}-2x+1+x^{3}+x\left(2x^{2}-2x+1\right)\left(-1\right)$$

Use the distributive property to multiply f by 2x^{2}-2x+1.

$$\left(2fx^{2}-2fx+f\right)\left(1-x\right)=2x^{2}-2x+1+x^{3}+x\left(2x^{2}-2x+1\right)\left(-1\right)$$

Use the distributive property to multiply 2fx^{2}-2fx+f by 1-x and combine like terms.

$$4fx^{2}-2fx^{3}-3fx+f=2x^{2}-2x+1+x^{3}+x\left(2x^{2}-2x+1\right)\left(-1\right)$$

Use the distributive property to multiply x by 2x^{2}-2x+1.

$$4fx^{2}-2fx^{3}-3fx+f=2x^{2}-2x+1+x^{3}+\left(2x^{3}-2x^{2}+x\right)\left(-1\right)$$

Use the distributive property to multiply 2x^{3}-2x^{2}+x by -1.

$$4fx^{2}-2fx^{3}-3fx+f=2x^{2}-2x+1+x^{3}-2x^{3}+2x^{2}-x$$

Combine x^{3} and -2x^{3} to get -x^{3}.

$$4fx^{2}-2fx^{3}-3fx+f=2x^{2}-2x+1-x^{3}+2x^{2}-x$$

Combine 2x^{2} and 2x^{2} to get 4x^{2}.

$$4fx^{2}-2fx^{3}-3fx+f=4x^{2}-2x+1-x^{3}-x$$

Combine -2x and -x to get -3x.

$$4fx^{2}-2fx^{3}-3fx+f=4x^{2}-3x+1-x^{3}$$

Combine all terms containing f.

$$\left(4x^{2}-2x^{3}-3x+1\right)f=4x^{2}-3x+1-x^{3}$$

The equation is in standard form.

$$\left(1-3x+4x^{2}-2x^{3}\right)f=1-3x+4x^{2}-x^{3}$$

Divide both sides by -2x^{3}+4x^{2}-3x+1.

$$\frac{\left(1-3x+4x^{2}-2x^{3}\right)f}{1-3x+4x^{2}-2x^{3}}=\frac{1-3x+4x^{2}-x^{3}}{1-3x+4x^{2}-2x^{3}}$$

Dividing by -2x^{3}+4x^{2}-3x+1 undoes the multiplication by -2x^{3}+4x^{2}-3x+1.

$$f=\frac{1-3x+4x^{2}-x^{3}}{1-3x+4x^{2}-2x^{3}}$$

Divide 4x^{2}-3x-x^{3}+1 by -2x^{3}+4x^{2}-3x+1.

$$f=\frac{1-3x+4x^{2}-x^{3}}{\left(x-1\right)\left(-2x^{2}+2x-1\right)}$$