Solve f(x+1/x)=x^2+1/x^2+6=f(x)

$$f ( x + \frac { 1 } { x } ) = x ^ { 2 } + \frac { 1 } { x ^ { 2 } } + 6 = f ( x )$$

Solve for f

$$f=x+\frac{6}{x}+\frac{1}{x^{3}}$$

$$x\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x^{2}.

$$x^{2}x^{2}+1+x^{2}\times 6=fxx^{2}$$

To multiply powers of the same base, add their exponents. Add 2 and 2 to get 4.

$$x^{4}+1+x^{2}\times 6=fxx^{2}$$

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

$$x^{4}+1+x^{2}\times 6=fx^{3}$$

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

$$fx^{3}=x^{4}+1+x^{2}\times 6$$

The equation is in standard form.

$$x^{3}f=x^{4}+6x^{2}+1$$

Divide both sides by x^{3}.

$$\frac{x^{3}f}{x^{3}}=\frac{x^{4}+6x^{2}+1}{x^{3}}$$

Dividing by x^{3} undoes the multiplication by x^{3}.

$$f=\frac{x^{4}+6x^{2}+1}{x^{3}}$$

Divide x^{4}+1+6x^{2} by x^{3}.

$$f=x+\frac{6}{x}+\frac{1}{x^{3}}$$