What is the frequency response function for the Bode Plot pictured here?
Your Bode plot is a straight line. As we are on a semilog-scale, the equation is given by 

$$y = m \log(x) + b$$

In this case, $y$ would be the magnitude of the frequency response $|H(j\omega)|$ in $\mathrm{dB}$, and $x$ would be the frequency $\omega$. Taking two points of the plot and solving the $2\times2$ equation system should give you the function you're looking for.

As you didn't put the values of the $\omega$-axis, I'm going to assume that the plot starts at $\omega=1$. In this case, when $\omega = 1$ the magnitude is $0 \ \mathrm{dB}$, so:

$$0 = m \log(1) + b\implies b=0$$

Also, when $\omega = 10$, the magnitude is $20 \ \mathrm{dB}$:

$$20 = m \log(10) +0\implies m=20$$

So the function is:

$$|H(j\omega)|_{\mathrm{dB}}=20\log(\omega)$$

Note that if your plot doesn't begin at $\omega=1$ (what I just assumed to be true), the reasoning would be the same but $b$ would not be zero.