Solve 7f(x)+f(-x)=4x^3+7x

$$7 f ( x ) + f ( - x ) = 4 x ^ { 3 } + 7 x$$

Solve for f (complex solution)

$$\left\{\begin{matrix}\\f=\frac{2x^{2}}{3}+\frac{7}{6}\text{, }&\text{unconditionally}\\f\in \mathrm{C}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine 7fx and f\left(-1\right)x to get 6fx.

$$6fx=4x^{3}+7x$$

The equation is in standard form.

$$6xf=4x^{3}+7x$$

Divide both sides by 6x.

$$\frac{6xf}{6x}=\frac{x\left(4x^{2}+7\right)}{6x}$$

Dividing by 6x undoes the multiplication by 6x.

$$f=\frac{x\left(4x^{2}+7\right)}{6x}$$

Divide x\left(4x^{2}+7\right) by 6x.

$$f=\frac{2x^{2}}{3}+\frac{7}{6}$$

Solve for f

$$\left\{\begin{matrix}\\f=\frac{2x^{2}}{3}+\frac{7}{6}\text{, }&\text{unconditionally}\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

Combine 7fx and f\left(-1\right)x to get 6fx.

$$6fx=4x^{3}+7x$$

The equation is in standard form.

$$6xf=4x^{3}+7x$$

Divide both sides by 6x.

$$\frac{6xf}{6x}=\frac{x\left(4x^{2}+7\right)}{6x}$$

Dividing by 6x undoes the multiplication by 6x.

$$f=\frac{x\left(4x^{2}+7\right)}{6x}$$

Divide x\left(4x^{2}+7\right) by 6x.

$$f=\frac{2x^{2}}{3}+\frac{7}{6}$$

Solve for x (complex solution)

$$x=\frac{\sqrt{6f-7}}{2}$$

$$x=0$$

$$x=-\frac{\sqrt{6f-7}}{2}$$

Solve for x

$$\left\{\begin{matrix}\\x=0\text{, }&\text{unconditionally}\\x=\frac{\sqrt{6f-7}}{2}\text{; }x=-\frac{\sqrt{6f-7}}{2}\text{, }&f\geq \frac{7}{6}\end{matrix}\right.$$