Solve f^-1=-sqrt{x-1}

$$f ^ { - 1 } = - \sqrt { x - 1 }$$

Solve for f (complex solution)

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

$$x\neq 1$$

Steps for Solving Linear Equation

Reorder the terms.

$$\frac{1}{f}=-\sqrt{x-1}$$

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

$$1=-\sqrt{x-1}f$$

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

$$-\sqrt{x-1}f=1$$

The equation is in standard form.

$$\left(-\sqrt{x-1}\right)f=1$$

Divide both sides by -\sqrt{x-1}.

$$\frac{\left(-\sqrt{x-1}\right)f}{-\sqrt{x-1}}=\frac{1}{-\sqrt{x-1}}$$

Dividing by -\sqrt{x-1} undoes the multiplication by -\sqrt{x-1}.

$$f=\frac{1}{-\sqrt{x-1}}$$

Divide 1 by -\sqrt{x-1}.

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

Variable f cannot be equal to 0.

$$f=-\left(x-1\right)^{-\frac{1}{2}}\text{, }f\neq 0$$

Solve for f

$$f=-\frac{1}{\sqrt{x-1}}$$

$$x>1$$

Steps for Solving Linear Equation

Reorder the terms.

$$\frac{1}{f}=-\sqrt{x-1}$$

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

$$1=-\sqrt{x-1}f$$

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

$$-\sqrt{x-1}f=1$$

The equation is in standard form.

$$\left(-\sqrt{x-1}\right)f=1$$

Divide both sides by -\sqrt{x-1}.

$$\frac{\left(-\sqrt{x-1}\right)f}{-\sqrt{x-1}}=\frac{1}{-\sqrt{x-1}}$$

Dividing by -\sqrt{x-1} undoes the multiplication by -\sqrt{x-1}.

$$f=\frac{1}{-\sqrt{x-1}}$$

Divide 1 by -\sqrt{x-1}.

$$f=-\frac{1}{\sqrt{x-1}}$$

Variable f cannot be equal to 0.

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

Solve for x

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

$$f<0$$

Steps for Solving Radical Equations

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

$$-\sqrt{x-1}=f^{-1}$$

Reorder the terms.

$$-\sqrt{x-1}=\frac{1}{f}$$

Multiply both sides of the equation by f.

$$-\sqrt{x-1}f=1$$

Divide both sides by -f.

$$\frac{\left(-f\right)\sqrt{x-1}}{-f}=\frac{1}{-f}$$

Dividing by -f undoes the multiplication by -f.

$$\sqrt{x-1}=\frac{1}{-f}$$

Divide 1 by -f.

$$\sqrt{x-1}=-\frac{1}{f}$$

Square both sides of the equation.

$$x-1=\frac{1}{f^{2}}$$

Add 1 to both sides of the equation.

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

Subtracting -1 from itself leaves 0.

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

Subtract -1 from \frac{1}{f^{2}}.

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

Solve for x (complex solution)

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

$$|-\pi +arg(\sqrt{\frac{1}{f^{2}}}f)|<\pi \text{ and }f\neq 0$$