Solve f^-1(x)=5e^x-1

$$f ^ { - 1 } ( x ) = 5 e ^ { x } - 1$$

Solve for f

$$f=-\frac{x}{1-5e^{x}}$$

$$x\neq 0\text{ and }x\neq -\ln(5)$$

Steps for Solving Linear Equation

Reorder the terms.

$$\frac{1}{f}x=5e^{x}-1$$

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by f.

$$1x=5e^{x}f+f\left(-1\right)$$

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

$$5e^{x}f+f\left(-1\right)=1x$$

Reorder the terms.

$$5fe^{x}-f=x$$

Combine all terms containing f.

$$\left(5e^{x}-1\right)f=x$$

Divide both sides by 5e^{x}-1.

$$\frac{\left(5e^{x}-1\right)f}{5e^{x}-1}=\frac{x}{5e^{x}-1}$$

Dividing by 5e^{x}-1 undoes the multiplication by 5e^{x}-1.

$$f=\frac{x}{5e^{x}-1}$$

Variable f cannot be equal to 0.

$$f=\frac{x}{5e^{x}-1}\text{, }f\neq 0$$