Starting with a function \[f(x)=2x+1\] defined for \(x\) as any real number, we can represent it as a function machine. We put \(x\) in at the left and get \(f(x)\) out at the right.

The inverse function, \(f^{-1}\), is such that if we put in \(f(x)\) we’ll get out \(x\). Fill in the boxes below so that if you put in \(2x+1\) at the right, you get \(x\) out at the left.

To find out what the inverse function is, use the same machine, put in \(x\) at the right and see what comes out at the left. The expression that comes out at the left is equal to \(f^{-1}(x)\).

If we had written \(y=2x+1\), then we would have
\[\begin{equation} y=f(x). \label{eq:f} \end{equation}\]
The second function machine above tells us that putting \(y\) into \(f^{-1}\) we get out \(x\), so
\[\begin{equation} x=f^{-1}(y). \label{eq:f-1} \end{equation}\]

In other words, \(f^{-1}(y)\) is what we get by rearranging \(\eqref{eq:f}\) to make \(x\) the subject. We can then simply replace each \(y\) with an \(x\) to find \(f^{-1}(x)\).

Complete the following.
\[\begin{align*}
y &= 2x+1 \\
\\
2x &= \\
\\
x &= \qquad\qquad\qquad = f^{-1}(y) \\
\\
\text{Therefore}\quad f^{-1}(x) &=
\end{align*}\]

Note we have chosen functions that can be written as simple function machines. There are many functions which cannot be written like this and many which cannot be inverted algebraically. Can you think of some examples?