Solve f^-1(x)=8-9x/7+8x

$$f ^ { - 1 } ( x ) = \frac { 8 - 9 x } { 7 + 8 x }$$

Solve for f

$$f=-\frac{x\left(8x+7\right)}{9x-8}$$

$$x\neq -\frac{7}{8}\text{ and }x\neq 0\text{ and }x\neq \frac{8}{9}$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 8x+7.

$$f^{-1}x\left(8x+7\right)=8-9x$$

Use the distributive property to multiply f^{-1}x by 8x+7.

$$8f^{-1}x^{2}+7f^{-1}x=8-9x$$

Reorder the terms.

$$8\times \frac{1}{f}x^{2}+7\times \frac{1}{f}x=-9x+8$$

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

$$8\times 1x^{2}+7\times 1x=-9xf+f\times 8$$

Do the multiplications.

$$8x^{2}+7x=-9xf+f\times 8$$

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

$$-9xf+f\times 8=8x^{2}+7x$$

Combine all terms containing f.

$$\left(-9x+8\right)f=8x^{2}+7x$$

The equation is in standard form.

$$\left(8-9x\right)f=8x^{2}+7x$$

Divide both sides by -9x+8.

$$\frac{\left(8-9x\right)f}{8-9x}=\frac{x\left(8x+7\right)}{8-9x}$$

Dividing by -9x+8 undoes the multiplication by -9x+8.

$$f=\frac{x\left(8x+7\right)}{8-9x}$$

Variable f cannot be equal to 0.

$$f=\frac{x\left(8x+7\right)}{8-9x}\text{, }f\neq 0$$

Solve for x (complex solution)

$$x=\frac{\sqrt{81f^{2}+382f+49}-9f-7}{16}$$

$$x=\frac{-\sqrt{81f^{2}+382f+49}-9f-7}{16}\text{, }f\neq 0$$

Solve for x

$$x=\frac{\sqrt{81f^{2}+382f+49}-9f-7}{16}$$

$$x=\frac{-\sqrt{81f^{2}+382f+49}-9f-7}{16}\text{, }f\leq \frac{-16\sqrt{127}-191}{81}\text{ or }\left(f\neq 0\text{ and }f\geq \frac{16\sqrt{127}-191}{81}\right)$$