To find the Inverse function of:

$\displaystyle f(x) = 3x + 21$

1. Replace $\displaystyle f(x)$ with $\displaystyle y$:

$\displaystyle y = 3x + 21$

2. Switch the variables $\displaystyle y$ and $\displaystyle x$:

$\displaystyle x = 3y + 21$

3. Solve for $\displaystyle y$:

$\displaystyle y = \frac{x - 21}{3}$

4. Simplify:

$\displaystyle y = \frac{1}{3}x - 7$

5. Replace $\displaystyle y$ with $\displaystyle f^{-1}(x)$ and the final solution is:

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

To find the Inverse function of:

$\displaystyle f(x) = 3x + 21$

1. Replace $\displaystyle f(x)$ with $\displaystyle y$:

$\displaystyle y = 3x + 21$

2. Switch the variables $\displaystyle y$ and $\displaystyle x$:

$\displaystyle x = 3y + 21$

3. Solve for $\displaystyle y$:

$\displaystyle y = \frac{x - 21}{3}$

4. Simplify:

$\displaystyle y = \frac{1}{3}x - 7$

5. Replace $\displaystyle y$ with $\displaystyle f^{-1}(x)$ and the final solution is:

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

To find an inverse of a function, switch x and y variables and solve:

$\displaystyle f^{-1}(x)=\frac{1}{5}x+2$

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

Subtract 2 from both sides:

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

Multiply both sides by 5:

$\displaystyle 5(x-2)=y$

Distribute:

$\displaystyle 5x-10=y$

The inverse is:

$\displaystyle f^{-1}(x)=5x-10$

To find the inverse function of the function given, we must replace all of the x's with y's and, vice versa:

$\displaystyle x=\frac{6y+1}{y}$

Note that in the problem statement we were given $\displaystyle f(x)$ and not $\displaystyle y$, but they mean the same thing in the sense that $\displaystyle y$ is a function of $\displaystyle x$.

Now, we just solve for y:

$\displaystyle xy=6y+1$

$\displaystyle xy-6y=1$

$\displaystyle y(x-6)=1$

$\displaystyle y=\frac{1}{x-6}$

To denote that this is the inverse of the original function, we can use the notation

$\displaystyle f^{-1}(x)=\frac{1}{x-6}$.

Before we begin, it would help to try and simplify the problem.  Let's start by factoring the numerator:

$\displaystyle x^2 + 4x + 3 = (x+1)(x+3)$

$\displaystyle f(x)=\frac{(x+1)(x+3)}{x + 1}$

We can see that the $\displaystyle (x+1)$ terms will cancel, making this problem much easier:

$\displaystyle f(x)= x+3$

To find the inverse, we can put a $\displaystyle y$ where the $\displaystyle x$ is currently, put a $\displaystyle x$ for the $\displaystyle f(x)$, and then solve for $\displaystyle y$.

$\displaystyle x = y + 3$

$\displaystyle y = x-3$

$\displaystyle f(x)^{-1}=x-3$

First we're going to change the $\displaystyle x$ in the function to a $\displaystyle y$, set the function equal to $\displaystyle x$, and solve for $\displaystyle y$:

$\displaystyle x=\frac{4}{y^2 + 3}$

Multiply each side by $\displaystyle y^2 + 3$:

$\displaystyle x(y^2 + 3)=4$

Divide each side by $\displaystyle x$:

$\displaystyle y^2 + 3=\frac{4}{x}$

Subtract $\displaystyle 3$ from each side:

$\displaystyle y^2 = \frac{4}{x}-3$

Finally, take the square root of each side:

$\displaystyle y=\pm\sqrt{\frac{4}{x}-3}$

To find the inverse of the function, we must first replace all x in the equation with y, and all y in the equation with x:

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

Now, solve for y:

$\displaystyle x(y+1)=y+3$

$\displaystyle xy+x=y+3$

$\displaystyle xy-y=3-x$

$\displaystyle y(x-1)=3-x$

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

Step 1: To find the inverse of a function $\displaystyle f(x)$, we will first switch the places of x and y. Wherever we saw y, we put x. Where we see x, we put y.

$\displaystyle y=x^3+1$

After changing places of letters:

$\displaystyle x=y^3+1$

Step 2: We want to solve for y, so we ned to get rid of the $\displaystyle 1$ on the right side.

$\displaystyle x-1=y^3$

Step 3: We want to find $\displaystyle y$, but we have $\displaystyle y^3$. To find $\displaystyle y$, we will take the cube root of both sides

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

This simplifies to:

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

The inverse function of $\displaystyle f(x)$ is $\displaystyle \sqrt[3]{x-1}=y$

To find the inverse of a function switch the position of x and y and solve to y.

$\displaystyle x=\frac{y-7}{8}\Rightarrow8x=y-7\Rightarrow8x+7=y$

Interchange the x and y-variables.

$\displaystyle x=4(y-6)$

Solve for y.  Distribute the constant through the binomial.

$\displaystyle x=4y-24$

Add 24 on both sides.

$\displaystyle x+24=4y-24+24$

The equation becomes:

$\displaystyle x+24=4y$

Divide by four on both sides.

$\displaystyle \frac{x+24}{4}=\frac{4y}{4}$

Simplify both sides.

The answer is:  $\displaystyle y=\frac{1}{4}x+6$