Solve 1:f(x)=1/1+5x

$$1 : f ( x ) = \frac { 1 } { 1 + 5 x }$$

Solve for f

$$f=x\left(5x+1\right)$$

$$x\neq -\frac{1}{5}\text{ and }x\neq 0$$

Solution Steps

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f\left(5x+1\right), the least common multiple of f,1+5x.

$$\left(5x+1\right)\times 1x=f$$

Use the distributive property to multiply 5x+1 by 1.

$$\left(5x+1\right)x=f$$

Use the distributive property to multiply 5x+1 by x.

$$5x^{2}+x=f$$

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

$$f=5x^{2}+x$$

Variable f cannot be equal to 0.

$$f=5x^{2}+x\text{, }f\neq 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{20f+1}-1}{10}$$

$$x=\frac{-\sqrt{20f+1}-1}{10}\text{, }f\neq 0$$

Solve for x

$$x=\frac{\sqrt{20f+1}-1}{10}$$

$$x=\frac{-\sqrt{20f+1}-1}{10}\text{, }f\neq 0\text{ and }f\geq -\frac{1}{20}$$