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

$$f ( x ) = x + 2 x ^ { 2 } \quad y \quad g ( x ) = \frac { 1 } { x }$$

Solve for g (complex solution)

$$\left\{\begin{matrix}g=-\frac{x^{2}-1}{2yx^{4}}\text{, }&x\neq 0\text{ and }y\neq 0\\g\in \mathrm{C}\text{, }&\left(x=-1\text{ or }x=1\right)\text{ and }y=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$xx+2x^{2}ygxx=1$$

Multiply x and x to get x^{2}.

$$x^{2}+2x^{2}ygxx=1$$

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

$$x^{2}+2x^{3}ygx=1$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$x^{2}+2x^{4}yg=1$$

Subtract x^{2} from both sides.

$$2x^{4}yg=1-x^{2}$$

The equation is in standard form.

$$2yx^{4}g=1-x^{2}$$

Divide both sides by 2x^{4}y.

$$\frac{2yx^{4}g}{2yx^{4}}=\frac{1-x^{2}}{2yx^{4}}$$

Dividing by 2x^{4}y undoes the multiplication by 2x^{4}y.

$$g=\frac{1-x^{2}}{2yx^{4}}$$

Solve for g

$$\left\{\begin{matrix}g=-\frac{x^{2}-1}{2yx^{4}}\text{, }&x\neq 0\text{ and }y\neq 0\\g\in \mathrm{R}\text{, }&y=0\text{ and }|x|=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Multiply both sides of the equation by x.

$$xx+2x^{2}ygxx=1$$

Multiply x and x to get x^{2}.

$$x^{2}+2x^{2}ygxx=1$$

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

$$x^{2}+2x^{3}ygx=1$$

To multiply powers of the same base, add their exponents. Add 3 and 1 to get 4.

$$x^{2}+2x^{4}yg=1$$

Subtract x^{2} from both sides.

$$2x^{4}yg=1-x^{2}$$

The equation is in standard form.

$$2yx^{4}g=1-x^{2}$$

Divide both sides by 2x^{4}y.

$$\frac{2yx^{4}g}{2yx^{4}}=\frac{1-x^{2}}{2yx^{4}}$$

Dividing by 2x^{4}y undoes the multiplication by 2x^{4}y.

$$g=\frac{1-x^{2}}{2yx^{4}}$$

Solve for x (complex solution)

$$\left\{\begin{matrix}x=-\frac{ig^{-\frac{1}{2}}y^{-\frac{1}{2}}\sqrt{\sqrt{8gy+1}+1}}{2}\text{; }x=\frac{ig^{-\frac{1}{2}}y^{-\frac{1}{2}}\sqrt{\sqrt{8gy+1}+1}}{2}\text{; }x=\frac{g^{-\frac{1}{2}}y^{-\frac{1}{2}}\sqrt{\sqrt{8gy+1}-1}}{2}\text{; }x=-\frac{g^{-\frac{1}{2}}y^{-\frac{1}{2}}\sqrt{\sqrt{8gy+1}-1}}{2}\text{, }&y\neq 0\text{ and }g\neq 0\\x=-1\text{; }x=1\text{, }&g=0\text{ or }y=0\end{matrix}\right.$$

Solve for x

$$\left\{\begin{matrix}x=\frac{\sqrt{-\frac{\sqrt{8gy+1}+1}{gy}}}{2}\text{; }x=-\frac{\sqrt{-\frac{\sqrt{8gy+1}+1}{gy}}}{2}\text{, }&\left(g>0\text{ and }y<0\text{ and }y\geq -\frac{1}{8g}\right)\text{ or }\left(y=-\frac{1}{8g}\text{ and }g\neq 0\right)\text{ or }\left(g<0\text{ and }y>0\text{ and }y\leq -\frac{1}{8g}\right)\\x=\frac{\sqrt{\frac{\sqrt{8gy+1}-1}{gy}}}{2}\text{; }x=-\frac{\sqrt{\frac{\sqrt{8gy+1}-1}{gy}}}{2}\text{, }&\left(y\leq -\frac{1}{8g}\text{ and }y>0\text{ and }g<0\right)\text{ or }\left(y\geq -\frac{1}{8g}\text{ and }y<0\text{ and }g>0\right)\text{ or }\left(y\neq 0\text{ and }y<-\frac{1}{8g}\text{ and }g<0\right)\text{ or }\left(y\neq 0\text{ and }y>-\frac{1}{8g}\text{ and }g>0\right)\text{ or }\left(y=-\frac{1}{8g}\text{ and }g\neq 0\right)\\x=1\text{; }x=-1\text{, }&g=0\text{ or }y=0\end{matrix}\right.$$