Interchange the x and y-variables, and solve for y.

\(\displaystyle x=\frac{1}{6}y+6\)

Subtract six from both sides.

\(\displaystyle x-6=\frac{1}{6}y+6-6\)

Simplify the right side.

\(\displaystyle x-6=\frac{1}{6}y\)

In order to isolate the y-variable, we will need to multiply six on both sides.

\(\displaystyle 6(x-6)=\frac{1}{6}y\cdot 6\)

Simplify both sides.  Distribute the integer through the binomial on the left.

\(\displaystyle 6x-36=y\)

The answer is:  \(\displaystyle y=6x-36\)

Interchange the x and y-variables.

\(\displaystyle x=4y+3\)

Solve for y.  Subtract three from both sides.

\(\displaystyle x-3=4y+3-3\)

Simplify the right side.

\(\displaystyle x-3=4y\)

Divide by four on both sides.

\(\displaystyle \frac{x-3}{4}=\frac{4y}{4}\)

Simplify both sides.

The answer is:  \(\displaystyle y=\frac{1}{4}x-\frac{3}{4}\)

Interchange the x and y-variables.  The equation becomes:

\(\displaystyle x=y^3+5\)

Subtract five from both sides.

\(\displaystyle x-5=y^3+5-5\)

\(\displaystyle x-5=y^3\)

Take the cubed root on both sides.  This will eliminate the cubed exponent.

\(\displaystyle \sqrt[3]{x-5}=\sqrt[3]{y^3}\)

The answer is:  \(\displaystyle y=\sqrt[3]{x-5}\)

Interchange the x and y-variables.

\(\displaystyle x=\frac{2}{5}y-\frac{1}{2}\)

Solve for y.  Add one-half on both sides.

\(\displaystyle x+\frac{1}{2}=\frac{2}{5}y-\frac{1}{2}+\frac{1}{2}\)

Simplify both sides.

\(\displaystyle x+\frac{1}{2}=\frac{2}{5}y\)

Multiply five over two on both sides in order to isolate the y-variable.

\(\displaystyle \frac{5}{2}(x+\frac{1}{2})=\frac{2}{5}y\cdot \frac{5}{2}\)

Apply the distributive property on the left side.  The right side will reduce to just a lone y-variable.

The answer is:  \(\displaystyle y=\frac{5}{2}x+\frac{5}{4}\)

Interchange the x and y-variables and solve for y.

\(\displaystyle x=-2y-1\)

Add one on both sides.

\(\displaystyle x+1=-2y-1+1\)

\(\displaystyle x+1=-2y\)

Divide by negative two on both sides.

\(\displaystyle \frac{x+1}{-2}=\frac{-2y}{-2}\)

Simplify the fractions.

The answer is:  \(\displaystyle y=-\frac{1}{2}x-\frac{1}{2}\)

In order to find the inverse of this function, interchange the x and y-variables.

\(\displaystyle x= 10y+3\)

Subtract three from both sides.

\(\displaystyle x-3= 10y+3-3\)

Simplify the equation.

\(\displaystyle x-3= 10y\)

Divide by ten on both sides.

\(\displaystyle \frac{x-3}{10}= \frac{10y}{10}\)

Simplify both sides.

The answer is:  \(\displaystyle y=\frac{1}{10}x-\frac{3}{10}\)

To find the inverse function, swap the x and y-variables.

\(\displaystyle x=-2y-30\)

Solve for y.  Add 30 on both sides.

\(\displaystyle x+30=-2y-30+30\)

Simplify the right side.

\(\displaystyle x+30=-2y\)

Divide by negative two on both sides.'

\(\displaystyle \frac{x+30}{-2}=\frac{-2y}{-2}\)

Simplify both fractions.

The answer is:  \(\displaystyle y=-\frac{1}{2}x-15\)

To determine the inverse function \(\displaystyle f^-^1\) of an explicitly defined function \(\displaystyle f\), substitute the dependent variable \(\displaystyle f(x)=y\) and independent variable \(\displaystyle x\) for \(\displaystyle x\) and \(\displaystyle y\) respectively, and then solve the resultant equation for \(\displaystyle y\). This new equation will define the inverse function \(\displaystyle f^-^1\), provided that \(\displaystyle f(f^-^1(x))=x\) and \(\displaystyle f^-^1(f(x))=x\) for every \(\displaystyle x\) in the domain of \(\displaystyle f\).

For this particular function, let \(\displaystyle y\) denote the dependent variable \(\displaystyle f(x)\):

\(\displaystyle y=\frac{7x+3}{2x-1}\).

Swap the variables \(\displaystyle y\) and \(\displaystyle x\):

\(\displaystyle x=\frac{7y+3}{2y-1}\).

Let us now solve this equation for \(\displaystyle y\). Multiplying both sides by \(\displaystyle 2y-1\) yields

\(\displaystyle 2yx-x=7y+3\).

Subtracting the term \(\displaystyle 7y\) and adding the term \(\displaystyle x\) to both sides yields

\(\displaystyle 2yx-7y=x+3\).

On the left-hand side of the equation, factor out \(\displaystyle y\) from both terms using the distributive property to yield

\(\displaystyle y(2x-7)=x+3\).

Now divide both sides of the equation by \(\displaystyle 2x-7\) to isolate the variable \(\displaystyle y\):

\(\displaystyle y=\frac{x+3}{2x-7}\).

In order to communicate the idea that this equation defines the inverse function to \(\displaystyle f\), let \(\displaystyle y=f^-^1(x)\) to yield the final answer:

\(\displaystyle f^-^1(x)=\frac{x+3}{2x-7}\).

To verify that this function is indeed the inverse of \(\displaystyle f\), calculate \(\displaystyle f(f^-^1(x))\) and \(\displaystyle f^-^1(f(x))\). 

\(\displaystyle f(f^-^1(x))=\frac{7(\frac{x+3}{2x-7})+3}{2(\frac{x+3}{2x-7})-1}\)

                     \(\displaystyle =\frac{\frac{7x+21+6x-21}{2x-7}}{\frac{2x+6-2x+7}{2x-7}}\)

                     \(\displaystyle =\frac{13x}{13}\)

                     \(\displaystyle =x\) ,

\(\displaystyle f^-^1(f(x))=\frac{(\frac{7x+3}{2x-1})+3}{2(\frac{7x+3}{2x-1})-7}\)

                    \(\displaystyle =\frac{\frac{7x+3+6x-3}{2x-1}}{\frac{14x+6-14x+7}{2x-1}}\)

                    \(\displaystyle =\frac{13x}{13}\)

                    \(\displaystyle =x\) .

Hence, the inverse function of the function \(\displaystyle f\) is

\(\displaystyle f^-^1(x)=\frac{x+3}{2x-7}\)

Interchange the x and y-variables and solve for y.

\(\displaystyle x=9(6y-2)\)

Distribute the nine through the binomial.

\(\displaystyle x=9(6y)+9(-2) = 54y-18\)

\(\displaystyle x= 54y-18\)

Add eighteen on both sides.

\(\displaystyle x+(18)= 54y-18+(18)\)

\(\displaystyle x+18=54y\)

Divide by 54 on both sides.

\(\displaystyle \frac{x+18}{54}=\frac{54y}{54}\)

The answer is:  \(\displaystyle y=\frac{1}{54}x+\frac{1}{3}\)

Interchange the x and y-variables and solve for y.

\(\displaystyle x=2(9-2y)\)

Distribute the integer through the binomial.

\(\displaystyle x=2(9-2y) = 18-4y\)

\(\displaystyle x= 18-4y\)

Add \(\displaystyle 4y\) on both sides.

\(\displaystyle x+(4y)= 18-4y+(4y)\)

\(\displaystyle x+4y=18\)

Subtract \(\displaystyle x\) on both sides.

\(\displaystyle 4y=-x+18\)

Divide by four on both sides.

\(\displaystyle \frac{4y}{4}=\frac{-x+18}{4}\)

Simplify both sides.

The inverse is:  \(\displaystyle y=-\frac{1}{4}x+\frac{9}{2}\)