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:  \(\displaystyle y=\frac{1}{4}x+6\)