In order to find the inverse, switch the x and y variables in the function then solve for y.

$\displaystyle 3x+1=y$

Switching variables we get,

 $\displaystyle 3y+1=x$.

 Then solving for y to get our final answer.

$\displaystyle 3y=x-1$

$\displaystyle y=\frac{x-1}{3}$

First, switch the variables making $\displaystyle y=x^2$ into $\displaystyle x=y^{2}$.

Then solve for y by taking the square root of both sides.

$\displaystyle y^2=x$

$\displaystyle \sqrt {y^2}=\pm \sqrt x$

$\displaystyle y=\pm \sqrt x$

To find the inverse in this case, we need to switch our x and y variables and then solve for y.

Therefore,

$\displaystyle y=\sqrt{\ln(x)+2}$ becomes,

$\displaystyle x=\sqrt{\ln(y)+2}$

To solve for y we square both sides to get rid of the sqaure root.

$\displaystyle x^2=\ln(y)+2$

We then subtract 2 from both sides and take the exponenetial of each side, leaving us with the final answer.

$\displaystyle x^2-2=\ln(y)$

$\displaystyle e^{x^2-2}=e^{\ln(y)}$

$\displaystyle e^x^{^{2}-2}=y$

To find the inverse of y, or 

$\displaystyle y^{-1}$

first switch your variables x and y in the equation. 

 $\displaystyle x_{i}=y_i^3+e^0$

Second, solve for the variable $\displaystyle y_i$ in the resulting equation. 

$\displaystyle x_i-e^0=y_i^3$

$\displaystyle (x_i-e^0)^\frac{1}{3}=(y_i^{3})^\frac{1}{3}$

$\displaystyle y_i=\sqrt[3]{x_i-e^0}$

Simplifying a number with 0 as the power, the inverse is

$\displaystyle y^{-1}=\sqrt[{3}]{x-1}$

To find the inverse of y, or 

$\displaystyle y^{-1}$

first switch your variables x and y in the equation. 

$\displaystyle x_i=\ln(y_i)+\pi$

Second, solve for the variable $\displaystyle y_i$ in the resulting equation. 

$\displaystyle x_i-\pi=\ln\,y_i$

And by setting each side of the equation as powers of base e,

$\displaystyle y^{-1}=e^{x-\pi}$

To find the inverse we need to switch the variables and then solve for y.

$\displaystyle y=x+5$

Switching the variables we get the following equation,

$\displaystyle x=y+5$.

Now solve for y.

$\displaystyle y^{-1}=x-5$

So we first replace every $\displaystyle y$ with an $\displaystyle x$ and every $\displaystyle x$ with a $\displaystyle y$.

Our resulting equation is:

$\displaystyle x=10y+9$

Now we simply solve for y.

Subtract 9 from both sides:

$\displaystyle x-9=10y+9-9$

$\displaystyle x-9=10y$

Now divide both sides by 10:

$\displaystyle \frac{x-9}{10}=\frac{10y}{10}$

$\displaystyle \frac{x-9}{10}=y$

The inverse of

$\displaystyle y=10x+9$

is

$\displaystyle y=\frac{x-9}{10}$

To find the inverse of a function we just switch the places of all $\displaystyle x$ and $\displaystyle y$ with eachother.

So

$\displaystyle y=9x$

turns into

$\displaystyle x=9y$

Now we solve for $\displaystyle y$

Divide both sides by $\displaystyle 9$

$\displaystyle \frac{x}{9}=\frac{9y}{9}$

$\displaystyle y=\frac{x}{9}$

We begin by taking $\displaystyle f(x)=-\frac{3x}{5}+11$ and changing the $\displaystyle f(x)$ to a $\displaystyle y$, giving us $\displaystyle y=-\frac{3x}{5}+11$.

Next, we switch all of our $\displaystyle x's$ and $\displaystyle y's$, giving us $\displaystyle x=-\frac{3y}{5}+11$.

Finally, we solve for $\displaystyle y$ by subtracting $\displaystyle 11$ from each side, multiplying each side by $\displaystyle 5$, and dividing each side by $\displaystyle -3$, leaving us with,

 $\displaystyle y=-\frac{5x}{3}+\frac{55}{3}$.

To find the inverse of the function, we switch the switch the $\displaystyle x$ and $\displaystyle y$ variables in the function.

$\displaystyle y = 10x-1$

Switching $\displaystyle x$ and $\displaystyle y$ gives

$\displaystyle x = 10y -1$

Then, solving for $\displaystyle y$ gives our answer:

$\displaystyle x + 1 = 10y$

$\displaystyle \frac{x-1}{10} = y$