Solve f^-1(x)=1/4(x+5)

$$f ^ { - 1 } ( x ) = \frac { 1 } { 4 } ( x + 5 )$$

Solve for f

$$f=\frac{4x}{x+5}$$

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

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{4} by x+5.

$$f^{-1}x=\frac{1}{4}x+\frac{5}{4}$$

Reorder the terms.

$$\frac{1}{f}x=\frac{1}{4}x+\frac{5}{4}$$

Variable f cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 4f, the least common multiple of f,4.

$$4\times 1x=\frac{1}{4}x\times 4f+4f\times \frac{5}{4}$$

Multiply 4 and 1 to get 4.

$$4x=\frac{1}{4}x\times 4f+4f\times \frac{5}{4}$$

Multiply \frac{1}{4} and 4 to get 1.

$$4x=xf+4f\times \frac{5}{4}$$

Multiply 4 and \frac{5}{4} to get 5.

$$4x=xf+5f$$

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

$$xf+5f=4x$$

Combine all terms containing f.

$$\left(x+5\right)f=4x$$

Divide both sides by x+5.

$$\frac{\left(x+5\right)f}{x+5}=\frac{4x}{x+5}$$

Dividing by x+5 undoes the multiplication by x+5.

$$f=\frac{4x}{x+5}$$

Variable f cannot be equal to 0.

$$f=\frac{4x}{x+5}\text{, }f\neq 0$$

Solve for x

$$x=\frac{5f}{4-f}$$

$$f\neq 4\text{ and }f\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply \frac{1}{4} by x+5.

$$f^{-1}x=\frac{1}{4}x+\frac{5}{4}$$

Subtract \frac{1}{4}x from both sides.

$$f^{-1}x-\frac{1}{4}x=\frac{5}{4}$$

Reorder the terms.

$$-\frac{1}{4}x+\frac{1}{f}x=\frac{5}{4}$$

Multiply both sides of the equation by 4f, the least common multiple of 4,f.

$$-\frac{1}{4}x\times 4f+4\times 1x=5f$$

Multiply -\frac{1}{4} and 4 to get -1.

$$-xf+4\times 1x=5f$$

Multiply 4 and 1 to get 4.

$$-xf+4x=5f$$

Combine all terms containing x.

$$\left(-f+4\right)x=5f$$

The equation is in standard form.

$$\left(4-f\right)x=5f$$

Divide both sides by 4-f.

$$\frac{\left(4-f\right)x}{4-f}=\frac{5f}{4-f}$$

Dividing by 4-f undoes the multiplication by 4-f.

$$x=\frac{5f}{4-f}$$