FIR Impulse response & transfer function
I agree with Matt's comment, however i suspect that whoever gave you the assignment wants you to write down an equation describing the transfer behaviour in time domain for an arbitrary input.

When $x[n] = \delta[n]$, the impulse response of the given system would be 

$$h[n] = -\delta[n] + 2\delta[n-1] - 2\delta[n-2] \ .$$

You can find it in the output $y[n]$ by setting the input $x[n]$ to

$$ x[n] = \delta[n] \triangleq \begin{cases}
1 \qquad \text{if } n = 0 \\
0 \qquad \text{if } n \ne 0 \\
\end{cases}$$

Then the output at the end of the diagram is $h[0]=-1$. The $z^{-1}$ elements act as delays of one sample, so the next outputs are $h[1]=2$ and $h[2]=-2$, respectively.

Now to find out what the equation does to general inputs in time domain, write the same thing down in terms of $x[n]$ instead of $x[n]=\delta[n]$.

For any given $x[n]$, the resulting $y[n]$ is produced by the current input times $-1$ and the prior two inputs multiplied with their factors $2$ and $-2$. You can then write down the general solution as

$$y[n] = -x[n] + 2x[n-1] -2x[n-2] \ .$$

Keep in mind however that the term "time domain transfer function" could be misunderstood by some, which is probably the reason your search did not yield any results.