Solve [f*g](x)=(sqrt{x})(x^2+1)

$$[ f \cdot g ] ( x ) = ( \sqrt { x } ) ( x ^ { 2 } + 1 )$$

Solve for f (complex solution)

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

Steps for Solving Linear Equation

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

$$fgx=\sqrt{x}x^{2}+\sqrt{x}$$

The equation is in standard form.

$$gxf=\sqrt{x}x^{2}+\sqrt{x}$$

Divide both sides by gx.

$$\frac{gxf}{gx}=\frac{x^{\frac{5}{2}}+\sqrt{x}}{gx}$$

Dividing by gx undoes the multiplication by gx.

$$f=\frac{x^{\frac{5}{2}}+\sqrt{x}}{gx}$$

Divide x^{\frac{5}{2}}+\sqrt{x} by gx.

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

Solve for g (complex solution)

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

Steps for Solving Linear Equation

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

$$fgx=\sqrt{x}x^{2}+\sqrt{x}$$

The equation is in standard form.

$$fxg=\sqrt{x}x^{2}+\sqrt{x}$$

Divide both sides by fx.

$$\frac{fxg}{fx}=\frac{x^{\frac{5}{2}}+\sqrt{x}}{fx}$$

Dividing by fx undoes the multiplication by fx.

$$g=\frac{x^{\frac{5}{2}}+\sqrt{x}}{fx}$$

Divide x^{\frac{5}{2}}+\sqrt{x} by fx.

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

Solve for f

$$\left\{\begin{matrix}f=\frac{x^{2}+1}{\sqrt{x}g}\text{, }&g\neq 0\text{ and }x>0\\f\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$fgx=\sqrt{x}x^{2}+\sqrt{x}$$

The equation is in standard form.

$$gxf=\sqrt{x}x^{2}+\sqrt{x}$$

Divide both sides by gx.

$$\frac{gxf}{gx}=\frac{x^{\frac{5}{2}}+\sqrt{x}}{gx}$$

Dividing by gx undoes the multiplication by gx.

$$f=\frac{x^{\frac{5}{2}}+\sqrt{x}}{gx}$$

Divide x^{\frac{5}{2}}+\sqrt{x} by gx.

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

Solve for g

$$\left\{\begin{matrix}g=\frac{x^{2}+1}{\sqrt{x}f}\text{, }&f\neq 0\text{ and }x>0\\g\in \mathrm{R}\text{, }&x=0\end{matrix}\right.$$

Steps for Solving Linear Equation

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

$$fgx=\sqrt{x}x^{2}+\sqrt{x}$$

The equation is in standard form.

$$fxg=\sqrt{x}x^{2}+\sqrt{x}$$

Divide both sides by fx.

$$\frac{fxg}{fx}=\frac{x^{\frac{5}{2}}+\sqrt{x}}{fx}$$

Dividing by fx undoes the multiplication by fx.

$$g=\frac{x^{\frac{5}{2}}+\sqrt{x}}{fx}$$

Divide x^{\frac{5}{2}}+\sqrt{x} by fx.

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